FormalConjectures.ErdosProblems.«1049»

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import FormalConjecturesUtil

Erdős Problem 1049

References:

erdosproblems.com/1049

[Er48] Erdős, P., On arithmetical properties of Lambert series. J. Indian Math. Soc. (N.S.) (1948), 63-66.

namespace Erdos1049open ArithmeticFunction Filter

Let $t>1$ be a rational number. Is $\sum_{n=1}^\infty\frac{1}{t^n-1}=\sum_{n=1}^\infty \frac{\tau(n)}{t^n}$ irrational, where $\tau(n)$ counts the divisors of $n$?

A conjecture of Chowla.

@[category research open, AMS 11]
theorem erdos_1049 :
    answer(sorry) ↔ ∀ t : ℚ, t > 1 → Irrational (∑' n : ℕ+, 1 / ((t : ℝ) ^ (n : ℕ) - 1)) := by⊢ True ↔ ∀ t > 1, Irrational (∑' (n : ℕ+), 1 / (↑t ^ ↑n - 1))
  sorryAll goals completed! 🐙

Erdős [Er48] proved that this is true if $t\geq 2$ is an integer.

@[category research solved, AMS 11]
theorem erdos_1049.variants.geq_2_integer :
     ∀ t : ℤ, t ≥ 2 → Irrational (∑' n : ℕ+, 1 / ((t : ℝ) ^ (n : ℕ) - 1)) := by⊢ ∀ t ≥ 2, Irrational (∑' (n : ℕ+), 1 / (↑t ^ ↑n - 1))
  sorryAll goals completed! 🐙

Convergent case (|t| > 1).

Substitute r := t⁻¹ so ‖r‖ < 1, then apply Mathlib's series identity tsum_pow_div_one_sub_eq_tsum_sigma at k = 0: $$\sum_{n \ge 1} \frac{r^n}{1 - r^n} = \sum_{n \ge 1} \sigma_0(n) \cdot r^n.$$ After clearing denominators, both sides match the Lambert identity: LHS becomes 1/(t^n - 1) and RHS becomes τ(n) / t^n.


@[category API, AMS 11]
private lemma lambert_convergent (t : ℝ) (ht : 1 < |t|) :
    ∑' n : ℕ+, 1 / (t ^ (n : ℕ) - 1) =
    ∑' n : ℕ+, ((n : ℕ).divisors.card : ℝ) / (t ^ (n : ℕ)) := byt:ℝht:1 < |t|⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n
  -- `|t| > 1` implies `t ≠ 0`, hence `t^n ≠ 0` for all n.
  have ht0 : t ≠ 0 := fun h => byt:ℝht:1 < |t|h:t = 0⊢ Falset:ℝht:1 < |t|ht0:t ≠ 0⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n subst hht:1 < |0|⊢ Falset:ℝht:1 < |t|ht0:t ≠ 0⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n; simp at htht:1 < 0⊢ Falset:ℝht:1 < |t|ht0:t ≠ 0⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n; linarith [abs_nonneg (0:ℝ)]t:ℝht:1 < |t|ht0:t ≠ 0⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑nt:ℝht:1 < |t|ht0:t ≠ 0⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n
  have htn : ∀ n : ℕ, t ^ n ≠ 0 := fun n => pow_ne_zero n ht0t:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n
  -- Substitution `r := t⁻¹`, so `|r| < 1`.
  set r : ℝ := t⁻¹ with hr_deft:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n
  have hr_norm : ‖r‖ < 1 := by
    rw [Real.norm_eq_abs,t:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹⊢ |r| < 1t:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹⊢ |t|⁻¹ < 1t:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹hr_norm:‖r‖ < 1⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n hr_def,t:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹⊢ |t⁻¹| < 1t:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹⊢ |t|⁻¹ < 1t:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹hr_norm:‖r‖ < 1⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n abs_invt:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹⊢ |t|⁻¹ < 1t:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹⊢ |t|⁻¹ < 1t:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹hr_norm:‖r‖ < 1⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n]t:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹⊢ |t|⁻¹ < 1t:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹hr_norm:‖r‖ < 1⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n; exact inv_lt_one_of_one_lt₀ htt:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹hr_norm:‖r‖ < 1⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑nt:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹hr_norm:‖r‖ < 1⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n
  -- Apply the Mathlib identity. Now reduce each side of our goal to its form.
  have h := tsum_pow_div_one_sub_eq_tsum_sigma (k := 0) hr_normt:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹hr_norm:‖r‖ < 1h:∑' (n : ℕ+), ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n) = ∑' (n : ℕ+), ↑((sigma 0) ↑n) * r ^ ↑n⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n
  convert h using 1e'_2t:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹hr_norm:‖r‖ < 1h:∑' (n : ℕ+), ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n) = ∑' (n : ℕ+), ↑((sigma 0) ↑n) * r ^ ↑n⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n)e'_3t:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹hr_norm:‖r‖ < 1h:∑' (n : ℕ+), ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n) = ∑' (n : ℕ+), ↑((sigma 0) ↑n) * r ^ ↑n⊢ ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n = ∑' (n : ℕ+), ↑((sigma 0) ↑n) * r ^ ↑n
  -- LHS: show `1 / (t^n - 1) = r^n / (1 - r^n)`. After substituting `r = 1/t`,
  -- this is the algebraic identity `1/(t^n - 1) = (1/t^n) / (1 - 1/t^n)`,
  -- valid when `t^n ≠ 0` and `t^n ≠ 1`.
  ·e'_2t:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹hr_norm:‖r‖ < 1h:∑' (n : ℕ+), ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n) = ∑' (n : ℕ+), ↑((sigma 0) ↑n) * r ^ ↑n⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n) apply tsum_congre'_2t:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹hr_norm:‖r‖ < 1h:∑' (n : ℕ+), ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n) = ∑' (n : ℕ+), ↑((sigma 0) ↑n) * r ^ ↑n⊢ ∀ (b : ℕ+), 1 / (t ^ ↑b - 1) = ↑↑b ^ 0 * r ^ ↑b / (1 - r ^ ↑b); intro ne'_2t:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹hr_norm:‖r‖ < 1h:∑' (n : ℕ+), ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n) = ∑' (n : ℕ+), ↑((sigma 0) ↑n) * r ^ ↑nn:ℕ+⊢ 1 / (t ^ ↑n - 1) = ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n)
    have hp : t ^ (n : ℕ) ≠ 0 := htn ne'_2t:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹hr_norm:‖r‖ < 1h:∑' (n : ℕ+), ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n) = ∑' (n : ℕ+), ↑((sigma 0) ↑n) * r ^ ↑nn:ℕ+hp:t ^ ↑n ≠ 0⊢ 1 / (t ^ ↑n - 1) = ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n)
    have hrn : r ^ (n : ℕ) = (t ^ (n : ℕ))⁻¹ := byt:ℝht:1 < |t|⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑ne'_2t:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹hr_norm:‖r‖ < 1h:∑' (n : ℕ+), ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n) = ∑' (n : ℕ+), ↑((sigma 0) ↑n) * r ^ ↑nn:ℕ+hp:t ^ ↑n ≠ 0hrn:r ^ ↑n = (t ^ ↑n)⁻¹⊢ 1 / (t ^ ↑n - 1) = ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n) rw [hr_def,t:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹hr_norm:‖r‖ < 1h:∑' (n : ℕ+), ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n) = ∑' (n : ℕ+), ↑((sigma 0) ↑n) * r ^ ↑nn:ℕ+hp:t ^ ↑n ≠ 0⊢ t⁻¹ ^ ↑n = (t ^ ↑n)⁻¹e'_2t:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹hr_norm:‖r‖ < 1h:∑' (n : ℕ+), ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n) = ∑' (n : ℕ+), ↑((sigma 0) ↑n) * r ^ ↑nn:ℕ+hp:t ^ ↑n ≠ 0hrn:r ^ ↑n = (t ^ ↑n)⁻¹⊢ 1 / (t ^ ↑n - 1) = ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n) inv_powt:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹hr_norm:‖r‖ < 1h:∑' (n : ℕ+), ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n) = ∑' (n : ℕ+), ↑((sigma 0) ↑n) * r ^ ↑nn:ℕ+hp:t ^ ↑n ≠ 0⊢ (t ^ ↑n)⁻¹ = (t ^ ↑n)⁻¹e'_2t:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹hr_norm:‖r‖ < 1h:∑' (n : ℕ+), ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n) = ∑' (n : ℕ+), ↑((sigma 0) ↑n) * r ^ ↑nn:ℕ+hp:t ^ ↑n ≠ 0hrn:r ^ ↑n = (t ^ ↑n)⁻¹⊢ 1 / (t ^ ↑n - 1) = ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n)]e'_2t:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹hr_norm:‖r‖ < 1h:∑' (n : ℕ+), ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n) = ∑' (n : ℕ+), ↑((sigma 0) ↑n) * r ^ ↑nn:ℕ+hp:t ^ ↑n ≠ 0hrn:r ^ ↑n = (t ^ ↑n)⁻¹⊢ 1 / (t ^ ↑n - 1) = ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n)e'_2t:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹hr_norm:‖r‖ < 1h:∑' (n : ℕ+), ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n) = ∑' (n : ℕ+), ↑((sigma 0) ↑n) * r ^ ↑nn:ℕ+hp:t ^ ↑n ≠ 0hrn:r ^ ↑n = (t ^ ↑n)⁻¹⊢ 1 / (t ^ ↑n - 1) = ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n)
    -- `t^n ≠ 1`: would imply `|t|^n = 1`, but `|t| > 1` gives `|t|^n > 1` since `n ≥ 1`.
    have hne1 : t ^ (n : ℕ) - 1 ≠ 0 := byt:ℝht:1 < |t|⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑ne'_2t:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹hr_norm:‖r‖ < 1h:∑' (n : ℕ+), ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n) = ∑' (n : ℕ+), ↑((sigma 0) ↑n) * r ^ ↑nn:ℕ+hp:t ^ ↑n ≠ 0hrn:r ^ ↑n = (t ^ ↑n)⁻¹hne1:t ^ ↑n - 1 ≠ 0⊢ 1 / (t ^ ↑n - 1) = ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n)
      intro hct:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹hr_norm:‖r‖ < 1h:∑' (n : ℕ+), ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n) = ∑' (n : ℕ+), ↑((sigma 0) ↑n) * r ^ ↑nn:ℕ+hp:t ^ ↑n ≠ 0hrn:r ^ ↑n = (t ^ ↑n)⁻¹hc:t ^ ↑n - 1 = 0⊢ Falsee'_2t:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹hr_norm:‖r‖ < 1h:∑' (n : ℕ+), ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n) = ∑' (n : ℕ+), ↑((sigma 0) ↑n) * r ^ ↑nn:ℕ+hp:t ^ ↑n ≠ 0hrn:r ^ ↑n = (t ^ ↑n)⁻¹hne1:t ^ ↑n - 1 ≠ 0⊢ 1 / (t ^ ↑n - 1) = ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n)
      have ht1 : t ^ (n : ℕ) = 1 := byt:ℝht:1 < |t|⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑nt:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹hr_norm:‖r‖ < 1h:∑' (n : ℕ+), ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n) = ∑' (n : ℕ+), ↑((sigma 0) ↑n) * r ^ ↑nn:ℕ+hp:t ^ ↑n ≠ 0hrn:r ^ ↑n = (t ^ ↑n)⁻¹hc:t ^ ↑n - 1 = 0ht1:t ^ ↑n = 1⊢ Falsee'_2t:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹hr_norm:‖r‖ < 1h:∑' (n : ℕ+), ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n) = ∑' (n : ℕ+), ↑((sigma 0) ↑n) * r ^ ↑nn:ℕ+hp:t ^ ↑n ≠ 0hrn:r ^ ↑n = (t ^ ↑n)⁻¹hne1:t ^ ↑n - 1 ≠ 0⊢ 1 / (t ^ ↑n - 1) = ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n) linarith [sub_eq_zero.mp hc]t:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹hr_norm:‖r‖ < 1h:∑' (n : ℕ+), ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n) = ∑' (n : ℕ+), ↑((sigma 0) ↑n) * r ^ ↑nn:ℕ+hp:t ^ ↑n ≠ 0hrn:r ^ ↑n = (t ^ ↑n)⁻¹hc:t ^ ↑n - 1 = 0ht1:t ^ ↑n = 1⊢ Falsee'_2t:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹hr_norm:‖r‖ < 1h:∑' (n : ℕ+), ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n) = ∑' (n : ℕ+), ↑((sigma 0) ↑n) * r ^ ↑nn:ℕ+hp:t ^ ↑n ≠ 0hrn:r ^ ↑n = (t ^ ↑n)⁻¹hne1:t ^ ↑n - 1 ≠ 0⊢ 1 / (t ^ ↑n - 1) = ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n)t:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹hr_norm:‖r‖ < 1h:∑' (n : ℕ+), ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n) = ∑' (n : ℕ+), ↑((sigma 0) ↑n) * r ^ ↑nn:ℕ+hp:t ^ ↑n ≠ 0hrn:r ^ ↑n = (t ^ ↑n)⁻¹hc:t ^ ↑n - 1 = 0ht1:t ^ ↑n = 1⊢ Falsee'_2t:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹hr_norm:‖r‖ < 1h:∑' (n : ℕ+), ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n) = ∑' (n : ℕ+), ↑((sigma 0) ↑n) * r ^ ↑nn:ℕ+hp:t ^ ↑n ≠ 0hrn:r ^ ↑n = (t ^ ↑n)⁻¹hne1:t ^ ↑n - 1 ≠ 0⊢ 1 / (t ^ ↑n - 1) = ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n)
      have habs1 : |t| ^ (n : ℕ) = 1 := byt:ℝht:1 < |t|⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑nt:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹hr_norm:‖r‖ < 1h:∑' (n : ℕ+), ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n) = ∑' (n : ℕ+), ↑((sigma 0) ↑n) * r ^ ↑nn:ℕ+hp:t ^ ↑n ≠ 0hrn:r ^ ↑n = (t ^ ↑n)⁻¹hc:t ^ ↑n - 1 = 0ht1:t ^ ↑n = 1habs1:|t| ^ ↑n = 1⊢ Falsee'_2t:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹hr_norm:‖r‖ < 1h:∑' (n : ℕ+), ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n) = ∑' (n : ℕ+), ↑((sigma 0) ↑n) * r ^ ↑nn:ℕ+hp:t ^ ↑n ≠ 0hrn:r ^ ↑n = (t ^ ↑n)⁻¹hne1:t ^ ↑n - 1 ≠ 0⊢ 1 / (t ^ ↑n - 1) = ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n) rw [← abs_pow,t:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹hr_norm:‖r‖ < 1h:∑' (n : ℕ+), ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n) = ∑' (n : ℕ+), ↑((sigma 0) ↑n) * r ^ ↑nn:ℕ+hp:t ^ ↑n ≠ 0hrn:r ^ ↑n = (t ^ ↑n)⁻¹hc:t ^ ↑n - 1 = 0ht1:t ^ ↑n = 1⊢ |t ^ ↑n| = 1t:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹hr_norm:‖r‖ < 1h:∑' (n : ℕ+), ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n) = ∑' (n : ℕ+), ↑((sigma 0) ↑n) * r ^ ↑nn:ℕ+hp:t ^ ↑n ≠ 0hrn:r ^ ↑n = (t ^ ↑n)⁻¹hc:t ^ ↑n - 1 = 0ht1:t ^ ↑n = 1⊢ |1| = 1t:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹hr_norm:‖r‖ < 1h:∑' (n : ℕ+), ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n) = ∑' (n : ℕ+), ↑((sigma 0) ↑n) * r ^ ↑nn:ℕ+hp:t ^ ↑n ≠ 0hrn:r ^ ↑n = (t ^ ↑n)⁻¹hc:t ^ ↑n - 1 = 0ht1:t ^ ↑n = 1habs1:|t| ^ ↑n = 1⊢ Falsee'_2t:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹hr_norm:‖r‖ < 1h:∑' (n : ℕ+), ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n) = ∑' (n : ℕ+), ↑((sigma 0) ↑n) * r ^ ↑nn:ℕ+hp:t ^ ↑n ≠ 0hrn:r ^ ↑n = (t ^ ↑n)⁻¹hne1:t ^ ↑n - 1 ≠ 0⊢ 1 / (t ^ ↑n - 1) = ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n) ht1t:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹hr_norm:‖r‖ < 1h:∑' (n : ℕ+), ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n) = ∑' (n : ℕ+), ↑((sigma 0) ↑n) * r ^ ↑nn:ℕ+hp:t ^ ↑n ≠ 0hrn:r ^ ↑n = (t ^ ↑n)⁻¹hc:t ^ ↑n - 1 = 0ht1:t ^ ↑n = 1⊢ |1| = 1t:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹hr_norm:‖r‖ < 1h:∑' (n : ℕ+), ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n) = ∑' (n : ℕ+), ↑((sigma 0) ↑n) * r ^ ↑nn:ℕ+hp:t ^ ↑n ≠ 0hrn:r ^ ↑n = (t ^ ↑n)⁻¹hc:t ^ ↑n - 1 = 0ht1:t ^ ↑n = 1⊢ |1| = 1t:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹hr_norm:‖r‖ < 1h:∑' (n : ℕ+), ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n) = ∑' (n : ℕ+), ↑((sigma 0) ↑n) * r ^ ↑nn:ℕ+hp:t ^ ↑n ≠ 0hrn:r ^ ↑n = (t ^ ↑n)⁻¹hc:t ^ ↑n - 1 = 0ht1:t ^ ↑n = 1habs1:|t| ^ ↑n = 1⊢ Falsee'_2t:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹hr_norm:‖r‖ < 1h:∑' (n : ℕ+), ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n) = ∑' (n : ℕ+), ↑((sigma 0) ↑n) * r ^ ↑nn:ℕ+hp:t ^ ↑n ≠ 0hrn:r ^ ↑n = (t ^ ↑n)⁻¹hne1:t ^ ↑n - 1 ≠ 0⊢ 1 / (t ^ ↑n - 1) = ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n)]t:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹hr_norm:‖r‖ < 1h:∑' (n : ℕ+), ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n) = ∑' (n : ℕ+), ↑((sigma 0) ↑n) * r ^ ↑nn:ℕ+hp:t ^ ↑n ≠ 0hrn:r ^ ↑n = (t ^ ↑n)⁻¹hc:t ^ ↑n - 1 = 0ht1:t ^ ↑n = 1⊢ |1| = 1t:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹hr_norm:‖r‖ < 1h:∑' (n : ℕ+), ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n) = ∑' (n : ℕ+), ↑((sigma 0) ↑n) * r ^ ↑nn:ℕ+hp:t ^ ↑n ≠ 0hrn:r ^ ↑n = (t ^ ↑n)⁻¹hc:t ^ ↑n - 1 = 0ht1:t ^ ↑n = 1habs1:|t| ^ ↑n = 1⊢ Falsee'_2t:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹hr_norm:‖r‖ < 1h:∑' (n : ℕ+), ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n) = ∑' (n : ℕ+), ↑((sigma 0) ↑n) * r ^ ↑nn:ℕ+hp:t ^ ↑n ≠ 0hrn:r ^ ↑n = (t ^ ↑n)⁻¹hne1:t ^ ↑n - 1 ≠ 0⊢ 1 / (t ^ ↑n - 1) = ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n); simpt:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹hr_norm:‖r‖ < 1h:∑' (n : ℕ+), ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n) = ∑' (n : ℕ+), ↑((sigma 0) ↑n) * r ^ ↑nn:ℕ+hp:t ^ ↑n ≠ 0hrn:r ^ ↑n = (t ^ ↑n)⁻¹hc:t ^ ↑n - 1 = 0ht1:t ^ ↑n = 1habs1:|t| ^ ↑n = 1⊢ Falsee'_2t:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹hr_norm:‖r‖ < 1h:∑' (n : ℕ+), ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n) = ∑' (n : ℕ+), ↑((sigma 0) ↑n) * r ^ ↑nn:ℕ+hp:t ^ ↑n ≠ 0hrn:r ^ ↑n = (t ^ ↑n)⁻¹hne1:t ^ ↑n - 1 ≠ 0⊢ 1 / (t ^ ↑n - 1) = ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n)t:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹hr_norm:‖r‖ < 1h:∑' (n : ℕ+), ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n) = ∑' (n : ℕ+), ↑((sigma 0) ↑n) * r ^ ↑nn:ℕ+hp:t ^ ↑n ≠ 0hrn:r ^ ↑n = (t ^ ↑n)⁻¹hc:t ^ ↑n - 1 = 0ht1:t ^ ↑n = 1habs1:|t| ^ ↑n = 1⊢ Falsee'_2t:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹hr_norm:‖r‖ < 1h:∑' (n : ℕ+), ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n) = ∑' (n : ℕ+), ↑((sigma 0) ↑n) * r ^ ↑nn:ℕ+hp:t ^ ↑n ≠ 0hrn:r ^ ↑n = (t ^ ↑n)⁻¹hne1:t ^ ↑n - 1 ≠ 0⊢ 1 / (t ^ ↑n - 1) = ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n)
      have hlt : 1 < |t| ^ (n : ℕ) := one_lt_pow₀ ht n.pos.ne't:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹hr_norm:‖r‖ < 1h:∑' (n : ℕ+), ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n) = ∑' (n : ℕ+), ↑((sigma 0) ↑n) * r ^ ↑nn:ℕ+hp:t ^ ↑n ≠ 0hrn:r ^ ↑n = (t ^ ↑n)⁻¹hc:t ^ ↑n - 1 = 0ht1:t ^ ↑n = 1habs1:|t| ^ ↑n = 1hlt:1 < |t| ^ ↑n⊢ Falsee'_2t:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹hr_norm:‖r‖ < 1h:∑' (n : ℕ+), ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n) = ∑' (n : ℕ+), ↑((sigma 0) ↑n) * r ^ ↑nn:ℕ+hp:t ^ ↑n ≠ 0hrn:r ^ ↑n = (t ^ ↑n)⁻¹hne1:t ^ ↑n - 1 ≠ 0⊢ 1 / (t ^ ↑n - 1) = ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n)
      linarithe'_2t:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹hr_norm:‖r‖ < 1h:∑' (n : ℕ+), ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n) = ∑' (n : ℕ+), ↑((sigma 0) ↑n) * r ^ ↑nn:ℕ+hp:t ^ ↑n ≠ 0hrn:r ^ ↑n = (t ^ ↑n)⁻¹hne1:t ^ ↑n - 1 ≠ 0⊢ 1 / (t ^ ↑n - 1) = ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n)e'_2t:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹hr_norm:‖r‖ < 1h:∑' (n : ℕ+), ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n) = ∑' (n : ℕ+), ↑((sigma 0) ↑n) * r ^ ↑nn:ℕ+hp:t ^ ↑n ≠ 0hrn:r ^ ↑n = (t ^ ↑n)⁻¹hne1:t ^ ↑n - 1 ≠ 0⊢ 1 / (t ^ ↑n - 1) = ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n)
    rw [hrne'_2t:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹hr_norm:‖r‖ < 1h:∑' (n : ℕ+), ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n) = ∑' (n : ℕ+), ↑((sigma 0) ↑n) * r ^ ↑nn:ℕ+hp:t ^ ↑n ≠ 0hrn:r ^ ↑n = (t ^ ↑n)⁻¹hne1:t ^ ↑n - 1 ≠ 0⊢ 1 / (t ^ ↑n - 1) = ↑↑n ^ 0 * (t ^ ↑n)⁻¹ / (1 - (t ^ ↑n)⁻¹)e'_2t:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹hr_norm:‖r‖ < 1h:∑' (n : ℕ+), ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n) = ∑' (n : ℕ+), ↑((sigma 0) ↑n) * r ^ ↑nn:ℕ+hp:t ^ ↑n ≠ 0hrn:r ^ ↑n = (t ^ ↑n)⁻¹hne1:t ^ ↑n - 1 ≠ 0⊢ 1 / (t ^ ↑n - 1) = ↑↑n ^ 0 * (t ^ ↑n)⁻¹ / (1 - (t ^ ↑n)⁻¹)]e'_2t:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹hr_norm:‖r‖ < 1h:∑' (n : ℕ+), ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n) = ∑' (n : ℕ+), ↑((sigma 0) ↑n) * r ^ ↑nn:ℕ+hp:t ^ ↑n ≠ 0hrn:r ^ ↑n = (t ^ ↑n)⁻¹hne1:t ^ ↑n - 1 ≠ 0⊢ 1 / (t ^ ↑n - 1) = ↑↑n ^ 0 * (t ^ ↑n)⁻¹ / (1 - (t ^ ↑n)⁻¹); field_simpAll goals completed! 🐙
  -- RHS: `σ_0(n) · r^n = τ(n) / t^n` since `σ_0 = τ` and `r = 1/t`.
  ·e'_3t:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹hr_norm:‖r‖ < 1h:∑' (n : ℕ+), ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n) = ∑' (n : ℕ+), ↑((sigma 0) ↑n) * r ^ ↑n⊢ ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n = ∑' (n : ℕ+), ↑((sigma 0) ↑n) * r ^ ↑n apply tsum_congre'_3t:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹hr_norm:‖r‖ < 1h:∑' (n : ℕ+), ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n) = ∑' (n : ℕ+), ↑((sigma 0) ↑n) * r ^ ↑n⊢ ∀ (b : ℕ+), ↑(↑b).divisors.card / t ^ ↑b = ↑((sigma 0) ↑b) * r ^ ↑b; intro ne'_3t:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹hr_norm:‖r‖ < 1h:∑' (n : ℕ+), ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n) = ∑' (n : ℕ+), ↑((sigma 0) ↑n) * r ^ ↑nn:ℕ+⊢ ↑(↑n).divisors.card / t ^ ↑n = ↑((sigma 0) ↑n) * r ^ ↑n
    have hp : t ^ (n : ℕ) ≠ 0 := htn ne'_3t:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹hr_norm:‖r‖ < 1h:∑' (n : ℕ+), ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n) = ∑' (n : ℕ+), ↑((sigma 0) ↑n) * r ^ ↑nn:ℕ+hp:t ^ ↑n ≠ 0⊢ ↑(↑n).divisors.card / t ^ ↑n = ↑((sigma 0) ↑n) * r ^ ↑n
    have hrn : r ^ (n : ℕ) = (t ^ (n : ℕ))⁻¹ := byt:ℝht:1 < |t|⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑ne'_3t:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹hr_norm:‖r‖ < 1h:∑' (n : ℕ+), ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n) = ∑' (n : ℕ+), ↑((sigma 0) ↑n) * r ^ ↑nn:ℕ+hp:t ^ ↑n ≠ 0hrn:r ^ ↑n = (t ^ ↑n)⁻¹⊢ ↑(↑n).divisors.card / t ^ ↑n = ↑((sigma 0) ↑n) * r ^ ↑n rw [hr_def,t:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹hr_norm:‖r‖ < 1h:∑' (n : ℕ+), ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n) = ∑' (n : ℕ+), ↑((sigma 0) ↑n) * r ^ ↑nn:ℕ+hp:t ^ ↑n ≠ 0⊢ t⁻¹ ^ ↑n = (t ^ ↑n)⁻¹e'_3t:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹hr_norm:‖r‖ < 1h:∑' (n : ℕ+), ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n) = ∑' (n : ℕ+), ↑((sigma 0) ↑n) * r ^ ↑nn:ℕ+hp:t ^ ↑n ≠ 0hrn:r ^ ↑n = (t ^ ↑n)⁻¹⊢ ↑(↑n).divisors.card / t ^ ↑n = ↑((sigma 0) ↑n) * r ^ ↑n inv_powt:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹hr_norm:‖r‖ < 1h:∑' (n : ℕ+), ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n) = ∑' (n : ℕ+), ↑((sigma 0) ↑n) * r ^ ↑nn:ℕ+hp:t ^ ↑n ≠ 0⊢ (t ^ ↑n)⁻¹ = (t ^ ↑n)⁻¹e'_3t:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹hr_norm:‖r‖ < 1h:∑' (n : ℕ+), ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n) = ∑' (n : ℕ+), ↑((sigma 0) ↑n) * r ^ ↑nn:ℕ+hp:t ^ ↑n ≠ 0hrn:r ^ ↑n = (t ^ ↑n)⁻¹⊢ ↑(↑n).divisors.card / t ^ ↑n = ↑((sigma 0) ↑n) * r ^ ↑n]e'_3t:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹hr_norm:‖r‖ < 1h:∑' (n : ℕ+), ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n) = ∑' (n : ℕ+), ↑((sigma 0) ↑n) * r ^ ↑nn:ℕ+hp:t ^ ↑n ≠ 0hrn:r ^ ↑n = (t ^ ↑n)⁻¹⊢ ↑(↑n).divisors.card / t ^ ↑n = ↑((sigma 0) ↑n) * r ^ ↑ne'_3t:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹hr_norm:‖r‖ < 1h:∑' (n : ℕ+), ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n) = ∑' (n : ℕ+), ↑((sigma 0) ↑n) * r ^ ↑nn:ℕ+hp:t ^ ↑n ≠ 0hrn:r ^ ↑n = (t ^ ↑n)⁻¹⊢ ↑(↑n).divisors.card / t ^ ↑n = ↑((sigma 0) ↑n) * r ^ ↑n
    rw [hrn,e'_3t:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹hr_norm:‖r‖ < 1h:∑' (n : ℕ+), ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n) = ∑' (n : ℕ+), ↑((sigma 0) ↑n) * r ^ ↑nn:ℕ+hp:t ^ ↑n ≠ 0hrn:r ^ ↑n = (t ^ ↑n)⁻¹⊢ ↑(↑n).divisors.card / t ^ ↑n = ↑((sigma 0) ↑n) * (t ^ ↑n)⁻¹e'_3t:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹hr_norm:‖r‖ < 1h:∑' (n : ℕ+), ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n) = ∑' (n : ℕ+), ↑((sigma 0) ↑n) * r ^ ↑nn:ℕ+hp:t ^ ↑n ≠ 0hrn:r ^ ↑n = (t ^ ↑n)⁻¹⊢ ↑(↑n).divisors.card / t ^ ↑n = ↑(↑n).divisors.card * (t ^ ↑n)⁻¹ ArithmeticFunction.sigma_zero_applye'_3t:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹hr_norm:‖r‖ < 1h:∑' (n : ℕ+), ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n) = ∑' (n : ℕ+), ↑((sigma 0) ↑n) * r ^ ↑nn:ℕ+hp:t ^ ↑n ≠ 0hrn:r ^ ↑n = (t ^ ↑n)⁻¹⊢ ↑(↑n).divisors.card / t ^ ↑n = ↑(↑n).divisors.card * (t ^ ↑n)⁻¹e'_3t:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹hr_norm:‖r‖ < 1h:∑' (n : ℕ+), ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n) = ∑' (n : ℕ+), ↑((sigma 0) ↑n) * r ^ ↑nn:ℕ+hp:t ^ ↑n ≠ 0hrn:r ^ ↑n = (t ^ ↑n)⁻¹⊢ ↑(↑n).divisors.card / t ^ ↑n = ↑(↑n).divisors.card * (t ^ ↑n)⁻¹]e'_3t:ℝht:1 < |t|ht0:t ≠ 0htn:∀ (n : ℕ), t ^ n ≠ 0r:ℝ := t⁻¹hr_def:r = t⁻¹hr_norm:‖r‖ < 1h:∑' (n : ℕ+), ↑↑n ^ 0 * r ^ ↑n / (1 - r ^ ↑n) = ∑' (n : ℕ+), ↑((sigma 0) ↑n) * r ^ ↑nn:ℕ+hp:t ^ ↑n ≠ 0hrn:r ^ ↑n = (t ^ ↑n)⁻¹⊢ ↑(↑n).divisors.card / t ^ ↑n = ↑(↑n).divisors.card * (t ^ ↑n)⁻¹; field_simpAll goals completed! 🐙

Divergent case (|t| ≤ 1).

Both tsums equal 0 in this regime, but for different reasons in each sub-case. We split on t ∈ {1, 0, -1} and the generic |t| < 1, t ≠ 0 remainder, and use the same key non-summability lemma below to handle the cases where the series diverges.

t = 1: every LHS term is 1 / (1 - 1) = 0 (Lean convention), so the LHS sum is trivially 0. RHS is Σ τ(n), non-summable.

t = 0: every RHS term is τ(n) / 0 = 0 (Lean convention), so the RHS sum is trivially 0. LHS is Σ (-1), non-summable.

t = -1: alternating; LHS vanishes at even n but odd n give terms of magnitude 1/2, an infinite set. RHS terms have magnitude τ(n) ≥ 1.

|t| < 1, t ≠ 0: standard; bounded denominator gives lower-bounded reciprocal on LHS, and |t^n| ≤ 1 plus τ(n) ≥ 1 gives the RHS bound.

In every case, Lean's tsum_eq_zero_of_not_summable collapses the non- summable side to 0, matching the 0 on the other side.


@[category API, AMS 11]
private lemma lambert_divergent (t : ℝ) (ht : |t| ≤ 1) :
    ∑' n : ℕ+, 1 / (t ^ (n : ℕ) - 1) =
    ∑' n : ℕ+, ((n : ℕ).divisors.card : ℝ) / (t ^ (n : ℕ)) := byt:ℝht:|t| ≤ 1⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n
  -- `key`: a function with infinitely many terms bounded away from zero is
  -- not summable. Standard contrapositive of `Summable.tendsto_cofinite_zero`.
  have key : ∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c →
      Set.Infinite {n : ℕ+ | c ≤ |f n|} → ¬Summable f := by
    intro f c hc hinf hsumt:ℝht:|t| ≤ 1f:ℕ+ → ℝc:ℝhc:0 < chinf:{n | c ≤ |f n|}.Infinitehsum:Summable f⊢ Falset:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable f⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n
    have h := hsum.tendsto_cofinite_zerot:ℝht:|t| ≤ 1f:ℕ+ → ℝc:ℝhc:0 < chinf:{n | c ≤ |f n|}.Infinitehsum:Summable fh:Tendsto f cofinite (nhds 0)⊢ Falset:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable f⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n
    rw [Metric.tendsto_nhdst:ℝht:|t| ≤ 1f:ℕ+ → ℝc:ℝhc:0 < chinf:{n | c ≤ |f n|}.Infinitehsum:Summable fh:∀ ε > 0, ∀ᶠ (x : ℕ+) in cofinite, dist (f x) 0 < ε⊢ Falset:ℝht:|t| ≤ 1f:ℕ+ → ℝc:ℝhc:0 < chinf:{n | c ≤ |f n|}.Infinitehsum:Summable fh:∀ ε > 0, ∀ᶠ (x : ℕ+) in cofinite, dist (f x) 0 < ε⊢ Falset:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable f⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n] at ht:ℝht:|t| ≤ 1f:ℕ+ → ℝc:ℝhc:0 < chinf:{n | c ≤ |f n|}.Infinitehsum:Summable fh:∀ ε > 0, ∀ᶠ (x : ℕ+) in cofinite, dist (f x) 0 < ε⊢ Falset:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable f⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n
    have h1 := h c hct:ℝht:|t| ≤ 1f:ℕ+ → ℝc:ℝhc:0 < chinf:{n | c ≤ |f n|}.Infinitehsum:Summable fh:∀ ε > 0, ∀ᶠ (x : ℕ+) in cofinite, dist (f x) 0 < εh1:∀ᶠ (x : ℕ+) in cofinite, dist (f x) 0 < c⊢ Falset:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable f⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n
    rw [Filter.eventually_cofinitet:ℝht:|t| ≤ 1f:ℕ+ → ℝc:ℝhc:0 < chinf:{n | c ≤ |f n|}.Infinitehsum:Summable fh:∀ ε > 0, ∀ᶠ (x : ℕ+) in cofinite, dist (f x) 0 < εh1:{x | ¬dist (f x) 0 < c}.Finite⊢ Falset:ℝht:|t| ≤ 1f:ℕ+ → ℝc:ℝhc:0 < chinf:{n | c ≤ |f n|}.Infinitehsum:Summable fh:∀ ε > 0, ∀ᶠ (x : ℕ+) in cofinite, dist (f x) 0 < εh1:{x | ¬dist (f x) 0 < c}.Finite⊢ Falset:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable f⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n] at h1t:ℝht:|t| ≤ 1f:ℕ+ → ℝc:ℝhc:0 < chinf:{n | c ≤ |f n|}.Infinitehsum:Summable fh:∀ ε > 0, ∀ᶠ (x : ℕ+) in cofinite, dist (f x) 0 < εh1:{x | ¬dist (f x) 0 < c}.Finite⊢ Falset:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable f⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n
    refine hinf (h1.subset fun n hn => ?_)t:ℝht:|t| ≤ 1f:ℕ+ → ℝc:ℝhc:0 < chinf:{n | c ≤ |f n|}.Infinitehsum:Summable fh:∀ ε > 0, ∀ᶠ (x : ℕ+) in cofinite, dist (f x) 0 < εh1:{x | ¬dist (f x) 0 < c}.Finiten:ℕ+hn:n ∈ {n | c ≤ |f n|}⊢ n ∈ {x | ¬dist (f x) 0 < c}t:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable f⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n
    simp only [Set.mem_ofPred_eq, Real.dist_eq, sub_zero, not_lt]t:ℝht:|t| ≤ 1f:ℕ+ → ℝc:ℝhc:0 < chinf:{n | c ≤ |f n|}.Infinitehsum:Summable fh:∀ ε > 0, ∀ᶠ (x : ℕ+) in cofinite, dist (f x) 0 < εh1:{x | ¬dist (f x) 0 < c}.Finiten:ℕ+hn:n ∈ {n | c ≤ |f n|}⊢ c ≤ |f n|t:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable f⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n
    exact hnt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable f⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑nt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable f⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n
  -- Number of divisors of `n ∈ ℕ+` is at least 1 (since `1 ∈ n.divisors`).
  have hcard_pos : ∀ (n : ℕ+), (1 : ℝ) ≤ ((n : ℕ).divisors.card : ℝ) := by
    intro nt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fn:ℕ+⊢ 1 ≤ ↑(↑n).divisors.cardt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.card⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n
    have : 0 < (n : ℕ).divisors.card := byt:ℝht:|t| ≤ 1⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑nt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fn:ℕ+this:0 < (↑n).divisors.card⊢ 1 ≤ ↑(↑n).divisors.cardt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.card⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n
      apply Finset.card_pos.mprt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fn:ℕ+⊢ (↑n).divisors.Nonemptyt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fn:ℕ+this:0 < (↑n).divisors.card⊢ 1 ≤ ↑(↑n).divisors.cardt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.card⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n
      exact ⟨1, Nat.one_mem_divisors.mpr n.2.ne'⟩t:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fn:ℕ+this:0 < (↑n).divisors.card⊢ 1 ≤ ↑(↑n).divisors.cardt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.card⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑nt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fn:ℕ+this:0 < (↑n).divisors.card⊢ 1 ≤ ↑(↑n).divisors.cardt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.card⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n
    exact_mod_cast thist:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.card⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑nt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.card⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n
  -- Case t = 1: LHS terms are 1/0 = 0 by Lean's convention, so LHS sum = 0.
  -- RHS terms are τ(n)/1 = τ(n) ≥ 1, so RHS is non-summable.
  by_cases ht1 : t = 1post:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:t = 1⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑nnegt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n
  ·post:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:t = 1⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n subst ht1poskey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|1| ≤ 1⊢ ∑' (n : ℕ+), 1 / (1 ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / 1 ^ ↑n
    have hLzero : ∀ n : ℕ+, (1 : ℝ) / ((1 : ℝ) ^ (n : ℕ) - 1) = 0 := byt:ℝht:|t| ≤ 1⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑nposkey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|1| ≤ 1hLzero:∀ (n : ℕ+), 1 / (1 ^ ↑n - 1) = 0⊢ ∑' (n : ℕ+), 1 / (1 ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / 1 ^ ↑n intro nkey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|1| ≤ 1n:ℕ+⊢ 1 / (1 ^ ↑n - 1) = 0poskey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|1| ≤ 1hLzero:∀ (n : ℕ+), 1 / (1 ^ ↑n - 1) = 0⊢ ∑' (n : ℕ+), 1 / (1 ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / 1 ^ ↑n; simpposkey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|1| ≤ 1hLzero:∀ (n : ℕ+), 1 / (1 ^ ↑n - 1) = 0⊢ ∑' (n : ℕ+), 1 / (1 ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / 1 ^ ↑nposkey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|1| ≤ 1hLzero:∀ (n : ℕ+), 1 / (1 ^ ↑n - 1) = 0⊢ ∑' (n : ℕ+), 1 / (1 ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / 1 ^ ↑n
    rw [tsum_congr hLzero,poskey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|1| ≤ 1hLzero:∀ (n : ℕ+), 1 / (1 ^ ↑n - 1) = 0⊢ ∑' (b : ℕ+), 0 = ∑' (n : ℕ+), ↑(↑n).divisors.card / 1 ^ ↑nposkey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|1| ≤ 1hLzero:∀ (n : ℕ+), 1 / (1 ^ ↑n - 1) = 0⊢ 0 = ∑' (n : ℕ+), ↑(↑n).divisors.card / 1 ^ ↑n tsum_zeroposkey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|1| ≤ 1hLzero:∀ (n : ℕ+), 1 / (1 ^ ↑n - 1) = 0⊢ 0 = ∑' (n : ℕ+), ↑(↑n).divisors.card / 1 ^ ↑nposkey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|1| ≤ 1hLzero:∀ (n : ℕ+), 1 / (1 ^ ↑n - 1) = 0⊢ 0 = ∑' (n : ℕ+), ↑(↑n).divisors.card / 1 ^ ↑n]poskey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|1| ≤ 1hLzero:∀ (n : ℕ+), 1 / (1 ^ ↑n - 1) = 0⊢ 0 = ∑' (n : ℕ+), ↑(↑n).divisors.card / 1 ^ ↑n
    symmposkey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|1| ≤ 1hLzero:∀ (n : ℕ+), 1 / (1 ^ ↑n - 1) = 0⊢ ∑' (n : ℕ+), ↑(↑n).divisors.card / 1 ^ ↑n = 0; apply tsum_eq_zero_of_not_summableposkey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|1| ≤ 1hLzero:∀ (n : ℕ+), 1 / (1 ^ ↑n - 1) = 0⊢ ¬Summable fun b ↦ ↑(↑b).divisors.card / 1 ^ ↑b
    apply key _ 1 (bykey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|1| ≤ 1hLzero:∀ (n : ℕ+), 1 / (1 ^ ↑n - 1) = 0⊢ 0 < 1 norm_numAll goals completed! 🐙)
    convert Set.infinite_univ (α := ℕ+)key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|1| ≤ 1hLzero:∀ (n : ℕ+), 1 / (1 ^ ↑n - 1) = 0⊢ {n | 1 ≤ |↑(↑n).divisors.card / 1 ^ ↑n|} = Set.univ
    ext nkey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|1| ≤ 1hLzero:∀ (n : ℕ+), 1 / (1 ^ ↑n - 1) = 0n:ℕ+⊢ n ∈ {n | 1 ≤ |↑(↑n).divisors.card / 1 ^ ↑n|} ↔ n ∈ Set.univ
    simp only [one_pow, div_one, Set.mem_ofPred_eq, Set.mem_univ, iff_true]key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|1| ≤ 1hLzero:∀ (n : ℕ+), 1 / (1 ^ ↑n - 1) = 0n:ℕ+⊢ 1 ≤ |↑(↑n).divisors.card|
    rw [abs_of_nonneg (bykey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|1| ≤ 1hLzero:∀ (n : ℕ+), 1 / (1 ^ ↑n - 1) = 0n:ℕ+⊢ 0 ≤ ↑(↑n).divisors.cardkey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|1| ≤ 1hLzero:∀ (n : ℕ+), 1 / (1 ^ ↑n - 1) = 0n:ℕ+⊢ 1 ≤ ↑(↑n).divisors.card positivityAll goals completed! 🐙key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|1| ≤ 1hLzero:∀ (n : ℕ+), 1 / (1 ^ ↑n - 1) = 0n:ℕ+⊢ 1 ≤ ↑(↑n).divisors.card)]key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|1| ≤ 1hLzero:∀ (n : ℕ+), 1 / (1 ^ ↑n - 1) = 0n:ℕ+⊢ 1 ≤ ↑(↑n).divisors.card; exact hcard_pos nAll goals completed! 🐙
  -- Case t = 0: RHS terms are τ(n)/0 = 0 by Lean's convention, so RHS sum = 0.
  -- LHS terms are 1/(0 - 1) = -1, so LHS is non-summable.
  by_cases ht0 : t = 0post:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:t = 0⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑nnegt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n
  ·post:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:t = 0⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n subst ht0poskey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|0| ≤ 1ht1:¬0 = 1⊢ ∑' (n : ℕ+), 1 / (0 ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / 0 ^ ↑n
    have hRzero : ∀ n : ℕ+, ((n : ℕ).divisors.card : ℝ) / ((0 : ℝ) ^ (n : ℕ)) = 0 := byt:ℝht:|t| ≤ 1⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑nposkey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|0| ≤ 1ht1:¬0 = 1hRzero:∀ (n : ℕ+), ↑(↑n).divisors.card / 0 ^ ↑n = 0⊢ ∑' (n : ℕ+), 1 / (0 ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / 0 ^ ↑n
      intro nkey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|0| ≤ 1ht1:¬0 = 1n:ℕ+⊢ ↑(↑n).divisors.card / 0 ^ ↑n = 0poskey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|0| ≤ 1ht1:¬0 = 1hRzero:∀ (n : ℕ+), ↑(↑n).divisors.card / 0 ^ ↑n = 0⊢ ∑' (n : ℕ+), 1 / (0 ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / 0 ^ ↑n; rw [zero_pow n.pos.ne'key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|0| ≤ 1ht1:¬0 = 1n:ℕ+⊢ ↑(↑n).divisors.card / 0 = 0key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|0| ≤ 1ht1:¬0 = 1n:ℕ+⊢ ↑(↑n).divisors.card / 0 = 0poskey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|0| ≤ 1ht1:¬0 = 1hRzero:∀ (n : ℕ+), ↑(↑n).divisors.card / 0 ^ ↑n = 0⊢ ∑' (n : ℕ+), 1 / (0 ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / 0 ^ ↑n]key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|0| ≤ 1ht1:¬0 = 1n:ℕ+⊢ ↑(↑n).divisors.card / 0 = 0poskey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|0| ≤ 1ht1:¬0 = 1hRzero:∀ (n : ℕ+), ↑(↑n).divisors.card / 0 ^ ↑n = 0⊢ ∑' (n : ℕ+), 1 / (0 ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / 0 ^ ↑n; simpposkey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|0| ≤ 1ht1:¬0 = 1hRzero:∀ (n : ℕ+), ↑(↑n).divisors.card / 0 ^ ↑n = 0⊢ ∑' (n : ℕ+), 1 / (0 ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / 0 ^ ↑nposkey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|0| ≤ 1ht1:¬0 = 1hRzero:∀ (n : ℕ+), ↑(↑n).divisors.card / 0 ^ ↑n = 0⊢ ∑' (n : ℕ+), 1 / (0 ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / 0 ^ ↑n
    rw [tsum_congr hRzero,poskey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|0| ≤ 1ht1:¬0 = 1hRzero:∀ (n : ℕ+), ↑(↑n).divisors.card / 0 ^ ↑n = 0⊢ ∑' (n : ℕ+), 1 / (0 ^ ↑n - 1) = ∑' (b : ℕ+), 0poskey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|0| ≤ 1ht1:¬0 = 1hRzero:∀ (n : ℕ+), ↑(↑n).divisors.card / 0 ^ ↑n = 0⊢ ∑' (n : ℕ+), 1 / (0 ^ ↑n - 1) = 0 tsum_zeroposkey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|0| ≤ 1ht1:¬0 = 1hRzero:∀ (n : ℕ+), ↑(↑n).divisors.card / 0 ^ ↑n = 0⊢ ∑' (n : ℕ+), 1 / (0 ^ ↑n - 1) = 0poskey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|0| ≤ 1ht1:¬0 = 1hRzero:∀ (n : ℕ+), ↑(↑n).divisors.card / 0 ^ ↑n = 0⊢ ∑' (n : ℕ+), 1 / (0 ^ ↑n - 1) = 0]poskey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|0| ≤ 1ht1:¬0 = 1hRzero:∀ (n : ℕ+), ↑(↑n).divisors.card / 0 ^ ↑n = 0⊢ ∑' (n : ℕ+), 1 / (0 ^ ↑n - 1) = 0
    apply tsum_eq_zero_of_not_summableposkey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|0| ≤ 1ht1:¬0 = 1hRzero:∀ (n : ℕ+), ↑(↑n).divisors.card / 0 ^ ↑n = 0⊢ ¬Summable fun b ↦ 1 / (0 ^ ↑b - 1)
    apply key _ 1 (bykey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|0| ≤ 1ht1:¬0 = 1hRzero:∀ (n : ℕ+), ↑(↑n).divisors.card / 0 ^ ↑n = 0⊢ 0 < 1 norm_numAll goals completed! 🐙)
    convert Set.infinite_univ (α := ℕ+)key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|0| ≤ 1ht1:¬0 = 1hRzero:∀ (n : ℕ+), ↑(↑n).divisors.card / 0 ^ ↑n = 0⊢ {n | 1 ≤ |1 / (0 ^ ↑n - 1)|} = Set.univ
    ext nkey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|0| ≤ 1ht1:¬0 = 1hRzero:∀ (n : ℕ+), ↑(↑n).divisors.card / 0 ^ ↑n = 0n:ℕ+⊢ n ∈ {n | 1 ≤ |1 / (0 ^ ↑n - 1)|} ↔ n ∈ Set.univ
    simp only [zero_pow n.pos.ne', zero_sub, Set.mem_ofPred_eq, Set.mem_univ, iff_true]key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|0| ≤ 1ht1:¬0 = 1hRzero:∀ (n : ℕ+), ↑(↑n).divisors.card / 0 ^ ↑n = 0n:ℕ+⊢ 1 ≤ |1 / -1|
    norm_numAll goals completed! 🐙
  -- Case t = -1: alternating signs make `1/(t^n - 1)` vanish at even n but
  -- equal -1/2 at odd n. The set of odd `n ∈ ℕ+` is infinite, which is enough
  -- to invoke `key` on the LHS. RHS magnitude is τ(n) ≥ 1 everywhere.
  by_cases htneg1 : t = -1post:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:t = -1⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑nnegt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n
  ·post:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:t = -1⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n subst htneg1poskey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0⊢ ∑' (n : ℕ+), 1 / ((-1) ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / (-1) ^ ↑n
    -- Construct the infinite set of odd positive naturals via the injection
    -- `k ↦ 2k + 1`, which lands in `ℕ+` and is always odd.
    have hinf_odd : Set.Infinite {n : ℕ+ | Odd (n : ℕ)} := byt:ℝht:|t| ≤ 1⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑nposkey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.Infinite⊢ ∑' (n : ℕ+), 1 / ((-1) ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / (-1) ^ ↑n
      apply Set.infinite_of_injective_forall_mem
        (f := fun k : ℕ => (⟨2 * k + 1, Nat.succ_pos _⟩ : ℕ+))hikey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0⊢ Function.Injective fun k ↦ ⟨2 * k + 1, ⋯⟩hfkey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0⊢ ∀ (x : ℕ), ⟨2 * x + 1, ⋯⟩ ∈ {n | Odd ↑n}poskey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.Infinite⊢ ∑' (n : ℕ+), 1 / ((-1) ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / (-1) ^ ↑n
      ·hikey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0⊢ Function.Injective fun k ↦ ⟨2 * k + 1, ⋯⟩poskey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.Infinite⊢ ∑' (n : ℕ+), 1 / ((-1) ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / (-1) ^ ↑n intro a b habhikey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0a:ℕb:ℕhab:(fun k ↦ ⟨2 * k + 1, ⋯⟩) a = (fun k ↦ ⟨2 * k + 1, ⋯⟩) b⊢ a = bposkey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.Infinite⊢ ∑' (n : ℕ+), 1 / ((-1) ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / (-1) ^ ↑n; rw [Subtype.mk.injEqhikey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0a:ℕb:ℕhab:2 * a + 1 = 2 * b + 1⊢ a = bhikey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0a:ℕb:ℕhab:2 * a + 1 = 2 * b + 1⊢ a = bposkey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.Infinite⊢ ∑' (n : ℕ+), 1 / ((-1) ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / (-1) ^ ↑n] at habhikey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0a:ℕb:ℕhab:2 * a + 1 = 2 * b + 1⊢ a = bposkey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.Infinite⊢ ∑' (n : ℕ+), 1 / ((-1) ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / (-1) ^ ↑n; omegaAll goals completed! 🐙poskey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.Infinite⊢ ∑' (n : ℕ+), 1 / ((-1) ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / (-1) ^ ↑n
      ·hfkey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0⊢ ∀ (x : ℕ), ⟨2 * x + 1, ⋯⟩ ∈ {n | Odd ↑n}poskey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.Infinite⊢ ∑' (n : ℕ+), 1 / ((-1) ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / (-1) ^ ↑n intro khfkey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0k:ℕ⊢ ⟨2 * k + 1, ⋯⟩ ∈ {n | Odd ↑n}poskey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.Infinite⊢ ∑' (n : ℕ+), 1 / ((-1) ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / (-1) ^ ↑n; show Odd (2 * k + 1)hfkey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0k:ℕ⊢ Odd (2 * k + 1)poskey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.Infinite⊢ ∑' (n : ℕ+), 1 / ((-1) ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / (-1) ^ ↑n; exact ⟨k, rfl⟩poskey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.Infinite⊢ ∑' (n : ℕ+), 1 / ((-1) ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / (-1) ^ ↑nposkey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.Infinite⊢ ∑' (n : ℕ+), 1 / ((-1) ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / (-1) ^ ↑n
    have hL : ¬ Summable (fun n : ℕ+ => 1 / (((-1 : ℝ)) ^ (n : ℕ) - 1)) := byt:ℝht:|t| ≤ 1⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑nposkey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.InfinitehL:¬Summable fun n ↦ 1 / ((-1) ^ ↑n - 1)⊢ ∑' (n : ℕ+), 1 / ((-1) ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / (-1) ^ ↑n
      apply key _ (1/2) (bykey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.Infinite⊢ 0 < 1 / 2poskey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.InfinitehL:¬Summable fun n ↦ 1 / ((-1) ^ ↑n - 1)⊢ ∑' (n : ℕ+), 1 / ((-1) ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / (-1) ^ ↑n norm_numAll goals completed! 🐙poskey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.InfinitehL:¬Summable fun n ↦ 1 / ((-1) ^ ↑n - 1)⊢ ∑' (n : ℕ+), 1 / ((-1) ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / (-1) ^ ↑n)
      apply hinf_odd.monokey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.Infinite⊢ {n | Odd ↑n} ⊆ {n | 1 / 2 ≤ |1 / ((-1) ^ ↑n - 1)|}poskey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.InfinitehL:¬Summable fun n ↦ 1 / ((-1) ^ ↑n - 1)⊢ ∑' (n : ℕ+), 1 / ((-1) ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / (-1) ^ ↑n
      intro n hnkey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.Infiniten:ℕ+hn:n ∈ {n | Odd ↑n}⊢ n ∈ {n | 1 / 2 ≤ |1 / ((-1) ^ ↑n - 1)|}poskey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.InfinitehL:¬Summable fun n ↦ 1 / ((-1) ^ ↑n - 1)⊢ ∑' (n : ℕ+), 1 / ((-1) ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / (-1) ^ ↑n
      -- For odd n: (-1)^n = -1, so 1/((-1)^n - 1) = 1/(-2), magnitude 1/2.
      show (1/2 : ℝ) ≤ |1 / ((-1 : ℝ) ^ (n : ℕ) - 1)|key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.Infiniten:ℕ+hn:n ∈ {n | Odd ↑n}⊢ 1 / 2 ≤ |1 / ((-1) ^ ↑n - 1)|poskey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.InfinitehL:¬Summable fun n ↦ 1 / ((-1) ^ ↑n - 1)⊢ ∑' (n : ℕ+), 1 / ((-1) ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / (-1) ^ ↑n
      rw [Odd.neg_one_pow hnkey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.Infiniten:ℕ+hn:n ∈ {n | Odd ↑n}⊢ 1 / 2 ≤ |1 / (-1 - 1)|key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.Infiniten:ℕ+hn:n ∈ {n | Odd ↑n}⊢ 1 / 2 ≤ |1 / (-1 - 1)|poskey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.InfinitehL:¬Summable fun n ↦ 1 / ((-1) ^ ↑n - 1)⊢ ∑' (n : ℕ+), 1 / ((-1) ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / (-1) ^ ↑n]key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.Infiniten:ℕ+hn:n ∈ {n | Odd ↑n}⊢ 1 / 2 ≤ |1 / (-1 - 1)|poskey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.InfinitehL:¬Summable fun n ↦ 1 / ((-1) ^ ↑n - 1)⊢ ∑' (n : ℕ+), 1 / ((-1) ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / (-1) ^ ↑n; norm_numposkey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.InfinitehL:¬Summable fun n ↦ 1 / ((-1) ^ ↑n - 1)⊢ ∑' (n : ℕ+), 1 / ((-1) ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / (-1) ^ ↑nposkey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.InfinitehL:¬Summable fun n ↦ 1 / ((-1) ^ ↑n - 1)⊢ ∑' (n : ℕ+), 1 / ((-1) ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / (-1) ^ ↑n
    have hR : ¬ Summable (fun n : ℕ+ => ((n : ℕ).divisors.card : ℝ) / ((-1 : ℝ) ^ (n : ℕ))) := byt:ℝht:|t| ≤ 1⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑nposkey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.InfinitehL:¬Summable fun n ↦ 1 / ((-1) ^ ↑n - 1)hR:¬Summable fun n ↦ ↑(↑n).divisors.card / (-1) ^ ↑n⊢ ∑' (n : ℕ+), 1 / ((-1) ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / (-1) ^ ↑n
      apply key _ 1 (bykey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.InfinitehL:¬Summable fun n ↦ 1 / ((-1) ^ ↑n - 1)⊢ 0 < 1poskey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.InfinitehL:¬Summable fun n ↦ 1 / ((-1) ^ ↑n - 1)hR:¬Summable fun n ↦ ↑(↑n).divisors.card / (-1) ^ ↑n⊢ ∑' (n : ℕ+), 1 / ((-1) ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / (-1) ^ ↑n norm_numAll goals completed! 🐙poskey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.InfinitehL:¬Summable fun n ↦ 1 / ((-1) ^ ↑n - 1)hR:¬Summable fun n ↦ ↑(↑n).divisors.card / (-1) ^ ↑n⊢ ∑' (n : ℕ+), 1 / ((-1) ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / (-1) ^ ↑n)
      convert Set.infinite_univ (α := ℕ+)key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.InfinitehL:¬Summable fun n ↦ 1 / ((-1) ^ ↑n - 1)⊢ {n | 1 ≤ |↑(↑n).divisors.card / (-1) ^ ↑n|} = Set.univposkey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.InfinitehL:¬Summable fun n ↦ 1 / ((-1) ^ ↑n - 1)hR:¬Summable fun n ↦ ↑(↑n).divisors.card / (-1) ^ ↑n⊢ ∑' (n : ℕ+), 1 / ((-1) ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / (-1) ^ ↑n
      ext nkey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.InfinitehL:¬Summable fun n ↦ 1 / ((-1) ^ ↑n - 1)n:ℕ+⊢ n ∈ {n | 1 ≤ |↑(↑n).divisors.card / (-1) ^ ↑n|} ↔ n ∈ Set.univposkey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.InfinitehL:¬Summable fun n ↦ 1 / ((-1) ^ ↑n - 1)hR:¬Summable fun n ↦ ↑(↑n).divisors.card / (-1) ^ ↑n⊢ ∑' (n : ℕ+), 1 / ((-1) ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / (-1) ^ ↑n
      simp only [Set.mem_ofPred_eq, Set.mem_univ, iff_true]key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.InfinitehL:¬Summable fun n ↦ 1 / ((-1) ^ ↑n - 1)n:ℕ+⊢ 1 ≤ |↑(↑n).divisors.card / (-1) ^ ↑n|poskey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.InfinitehL:¬Summable fun n ↦ 1 / ((-1) ^ ↑n - 1)hR:¬Summable fun n ↦ ↑(↑n).divisors.card / (-1) ^ ↑n⊢ ∑' (n : ℕ+), 1 / ((-1) ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / (-1) ^ ↑n
      -- |(-1)^n| = 1, so |τ(n) / (-1)^n| = τ(n) ≥ 1.
      rw [abs_div,key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.InfinitehL:¬Summable fun n ↦ 1 / ((-1) ^ ↑n - 1)n:ℕ+⊢ 1 ≤ |↑(↑n).divisors.card| / |(-1) ^ ↑n|key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.InfinitehL:¬Summable fun n ↦ 1 / ((-1) ^ ↑n - 1)n:ℕ+⊢ 1 ≤ |↑(↑n).divisors.card|poskey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.InfinitehL:¬Summable fun n ↦ 1 / ((-1) ^ ↑n - 1)hR:¬Summable fun n ↦ ↑(↑n).divisors.card / (-1) ^ ↑n⊢ ∑' (n : ℕ+), 1 / ((-1) ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / (-1) ^ ↑n abs_pow,key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.InfinitehL:¬Summable fun n ↦ 1 / ((-1) ^ ↑n - 1)n:ℕ+⊢ 1 ≤ |↑(↑n).divisors.card| / |(-1)| ^ ↑nkey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.InfinitehL:¬Summable fun n ↦ 1 / ((-1) ^ ↑n - 1)n:ℕ+⊢ 1 ≤ |↑(↑n).divisors.card|poskey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.InfinitehL:¬Summable fun n ↦ 1 / ((-1) ^ ↑n - 1)hR:¬Summable fun n ↦ ↑(↑n).divisors.card / (-1) ^ ↑n⊢ ∑' (n : ℕ+), 1 / ((-1) ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / (-1) ^ ↑n abs_neg,key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.InfinitehL:¬Summable fun n ↦ 1 / ((-1) ^ ↑n - 1)n:ℕ+⊢ 1 ≤ |↑(↑n).divisors.card| / |1| ^ ↑nkey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.InfinitehL:¬Summable fun n ↦ 1 / ((-1) ^ ↑n - 1)n:ℕ+⊢ 1 ≤ |↑(↑n).divisors.card|poskey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.InfinitehL:¬Summable fun n ↦ 1 / ((-1) ^ ↑n - 1)hR:¬Summable fun n ↦ ↑(↑n).divisors.card / (-1) ^ ↑n⊢ ∑' (n : ℕ+), 1 / ((-1) ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / (-1) ^ ↑n abs_one,key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.InfinitehL:¬Summable fun n ↦ 1 / ((-1) ^ ↑n - 1)n:ℕ+⊢ 1 ≤ |↑(↑n).divisors.card| / 1 ^ ↑nkey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.InfinitehL:¬Summable fun n ↦ 1 / ((-1) ^ ↑n - 1)n:ℕ+⊢ 1 ≤ |↑(↑n).divisors.card|poskey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.InfinitehL:¬Summable fun n ↦ 1 / ((-1) ^ ↑n - 1)hR:¬Summable fun n ↦ ↑(↑n).divisors.card / (-1) ^ ↑n⊢ ∑' (n : ℕ+), 1 / ((-1) ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / (-1) ^ ↑n one_pow,key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.InfinitehL:¬Summable fun n ↦ 1 / ((-1) ^ ↑n - 1)n:ℕ+⊢ 1 ≤ |↑(↑n).divisors.card| / 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.InfinitehL:¬Summable fun n ↦ 1 / ((-1) ^ ↑n - 1)n:ℕ+⊢ 1 ≤ |↑(↑n).divisors.card|poskey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.InfinitehL:¬Summable fun n ↦ 1 / ((-1) ^ ↑n - 1)hR:¬Summable fun n ↦ ↑(↑n).divisors.card / (-1) ^ ↑n⊢ ∑' (n : ℕ+), 1 / ((-1) ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / (-1) ^ ↑n div_onekey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.InfinitehL:¬Summable fun n ↦ 1 / ((-1) ^ ↑n - 1)n:ℕ+⊢ 1 ≤ |↑(↑n).divisors.card|key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.InfinitehL:¬Summable fun n ↦ 1 / ((-1) ^ ↑n - 1)n:ℕ+⊢ 1 ≤ |↑(↑n).divisors.card|poskey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.InfinitehL:¬Summable fun n ↦ 1 / ((-1) ^ ↑n - 1)hR:¬Summable fun n ↦ ↑(↑n).divisors.card / (-1) ^ ↑n⊢ ∑' (n : ℕ+), 1 / ((-1) ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / (-1) ^ ↑n]key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.InfinitehL:¬Summable fun n ↦ 1 / ((-1) ^ ↑n - 1)n:ℕ+⊢ 1 ≤ |↑(↑n).divisors.card|poskey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.InfinitehL:¬Summable fun n ↦ 1 / ((-1) ^ ↑n - 1)hR:¬Summable fun n ↦ ↑(↑n).divisors.card / (-1) ^ ↑n⊢ ∑' (n : ℕ+), 1 / ((-1) ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / (-1) ^ ↑n
      rw [abs_of_nonneg (bykey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.InfinitehL:¬Summable fun n ↦ 1 / ((-1) ^ ↑n - 1)n:ℕ+⊢ 0 ≤ ↑(↑n).divisors.cardkey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.InfinitehL:¬Summable fun n ↦ 1 / ((-1) ^ ↑n - 1)n:ℕ+⊢ 1 ≤ ↑(↑n).divisors.cardposkey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.InfinitehL:¬Summable fun n ↦ 1 / ((-1) ^ ↑n - 1)hR:¬Summable fun n ↦ ↑(↑n).divisors.card / (-1) ^ ↑n⊢ ∑' (n : ℕ+), 1 / ((-1) ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / (-1) ^ ↑n positivityAll goals completed! 🐙key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.InfinitehL:¬Summable fun n ↦ 1 / ((-1) ^ ↑n - 1)n:ℕ+⊢ 1 ≤ ↑(↑n).divisors.cardposkey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.InfinitehL:¬Summable fun n ↦ 1 / ((-1) ^ ↑n - 1)hR:¬Summable fun n ↦ ↑(↑n).divisors.card / (-1) ^ ↑n⊢ ∑' (n : ℕ+), 1 / ((-1) ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / (-1) ^ ↑n)]key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.InfinitehL:¬Summable fun n ↦ 1 / ((-1) ^ ↑n - 1)n:ℕ+⊢ 1 ≤ ↑(↑n).divisors.cardposkey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.InfinitehL:¬Summable fun n ↦ 1 / ((-1) ^ ↑n - 1)hR:¬Summable fun n ↦ ↑(↑n).divisors.card / (-1) ^ ↑n⊢ ∑' (n : ℕ+), 1 / ((-1) ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / (-1) ^ ↑n
      exact hcard_pos nposkey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.InfinitehL:¬Summable fun n ↦ 1 / ((-1) ^ ↑n - 1)hR:¬Summable fun n ↦ ↑(↑n).divisors.card / (-1) ^ ↑n⊢ ∑' (n : ℕ+), 1 / ((-1) ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / (-1) ^ ↑nposkey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.InfinitehL:¬Summable fun n ↦ 1 / ((-1) ^ ↑n - 1)hR:¬Summable fun n ↦ ↑(↑n).divisors.card / (-1) ^ ↑n⊢ ∑' (n : ℕ+), 1 / ((-1) ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / (-1) ^ ↑n
    rw [tsum_eq_zero_of_not_summable hL,poskey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.InfinitehL:¬Summable fun n ↦ 1 / ((-1) ^ ↑n - 1)hR:¬Summable fun n ↦ ↑(↑n).divisors.card / (-1) ^ ↑n⊢ 0 = ∑' (n : ℕ+), ↑(↑n).divisors.card / (-1) ^ ↑nAll goals completed! 🐙 tsum_eq_zero_of_not_summable hRposkey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0hinf_odd:{n | Odd ↑n}.InfinitehL:¬Summable fun n ↦ 1 / ((-1) ^ ↑n - 1)hR:¬Summable fun n ↦ ↑(↑n).divisors.card / (-1) ^ ↑n⊢ 0 = 0All goals completed! 🐙]All goals completed! 🐙
  -- Remaining case: |t| ≤ 1 with t ∉ {1, 0, -1}, equivalently |t| < 1 and t ≠ 0.
  ·negt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n -- First narrow `|t| ≤ 1` to `|t| < 1` using the case exclusions.
    have habs_lt : |t| < 1 := by
      rcases lt_or_eq_of_le ht with h | hinlt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1h:|t| < 1⊢ |t| < 1inrt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1h:|t| = 1⊢ |t| < 1negt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n
      ·inlt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1h:|t| < 1⊢ |t| < 1negt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n exact hAll goals completed! 🐙negt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n
      ·inrt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1h:|t| = 1⊢ |t| < 1negt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n exfalsoinrt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1h:|t| = 1⊢ Falsenegt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n
        rcases (abs_eq zero_le_one).mp h with rfl | rflinr.inlkey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|1| ≤ 1ht1:¬1 = 1ht0:¬1 = 0htneg1:¬1 = -1h:|1| = 1⊢ Falseinr.inrkey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0htneg1:¬-1 = -1h:|(-1)| = 1⊢ Falsenegt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n
        ·inr.inlkey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|1| ≤ 1ht1:¬1 = 1ht0:¬1 = 0htneg1:¬1 = -1h:|1| = 1⊢ Falsenegt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n exact ht1 rflAll goals completed! 🐙negt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n
        ·inr.inrkey:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht:|(-1)| ≤ 1ht1:¬-1 = 1ht0:¬-1 = 0htneg1:¬-1 = -1h:|(-1)| = 1⊢ Falsenegt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n exact htneg1 rflnegt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑nnegt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n
    have habs_pos : 0 < |t| := abs_pos.mpr ht0negt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n
    -- Since `|t| < 1`, `|t^n| ≤ 1` for all `n ∈ ℕ+`.
    have hbound : ∀ (n : ℕ+), |t ^ (n : ℕ)| ≤ 1 := by
      intro nt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|n:ℕ+⊢ |t ^ ↑n| ≤ 1negt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n; rw [abs_powt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|n:ℕ+⊢ |t| ^ ↑n ≤ 1t:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|n:ℕ+⊢ |t| ^ ↑n ≤ 1negt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n]t:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|n:ℕ+⊢ |t| ^ ↑n ≤ 1negt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n; exact pow_le_one₀ (abs_nonneg _) (le_of_lt habs_lt)negt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑nnegt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n
    -- Since `|t| < 1` strictly, `t^n ≠ 1` (else `|t|^n = 1` but `|t|^n ≤ |t| < 1`).
    have htn_ne_one : ∀ (n : ℕ+), t ^ (n : ℕ) ≠ 1 := by
      intro n hnt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1n:ℕ+hn:t ^ ↑n = 1⊢ Falsenegt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n
      have h1 : |t ^ (n : ℕ)| = 1 := byt:ℝht:|t| ≤ 1⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑nt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1n:ℕ+hn:t ^ ↑n = 1h1:|t ^ ↑n| = 1⊢ Falsenegt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n rw [hnt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1n:ℕ+hn:t ^ ↑n = 1⊢ |1| = 1t:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1n:ℕ+hn:t ^ ↑n = 1⊢ |1| = 1t:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1n:ℕ+hn:t ^ ↑n = 1h1:|t ^ ↑n| = 1⊢ Falsenegt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n]t:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1n:ℕ+hn:t ^ ↑n = 1⊢ |1| = 1t:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1n:ℕ+hn:t ^ ↑n = 1h1:|t ^ ↑n| = 1⊢ Falsenegt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n; exact abs_onet:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1n:ℕ+hn:t ^ ↑n = 1h1:|t ^ ↑n| = 1⊢ Falsenegt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑nt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1n:ℕ+hn:t ^ ↑n = 1h1:|t ^ ↑n| = 1⊢ Falsenegt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n
      rw [abs_powt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1n:ℕ+hn:t ^ ↑n = 1h1:|t| ^ ↑n = 1⊢ Falset:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1n:ℕ+hn:t ^ ↑n = 1h1:|t| ^ ↑n = 1⊢ Falsenegt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n] at h1t:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1n:ℕ+hn:t ^ ↑n = 1h1:|t| ^ ↑n = 1⊢ Falsenegt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n
      have hle : |t| ^ (n : ℕ) ≤ |t| := byt:ℝht:|t| ≤ 1⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑nt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1n:ℕ+hn:t ^ ↑n = 1h1:|t| ^ ↑n = 1hle:|t| ^ ↑n ≤ |t|⊢ Falsenegt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n
        exact pow_le_of_le_one (abs_nonneg _) (le_of_lt habs_lt) n.pos.ne't:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1n:ℕ+hn:t ^ ↑n = 1h1:|t| ^ ↑n = 1hle:|t| ^ ↑n ≤ |t|⊢ Falsenegt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑nt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1n:ℕ+hn:t ^ ↑n = 1h1:|t| ^ ↑n = 1hle:|t| ^ ↑n ≤ |t|⊢ Falsenegt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n
      linarithnegt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑nnegt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n
    have htn_ne_zero : ∀ (n : ℕ+), t ^ (n : ℕ) ≠ 0 := fun n => pow_ne_zero _ ht0negt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n
    -- LHS: `|t^n - 1| ≤ |t^n| + 1 ≤ 2`, so `|1 / (t^n - 1)| ≥ 1/2` everywhere.
    have hL : ¬ Summable (fun n : ℕ+ => 1 / (t ^ (n : ℕ) - 1)) := by
      apply key _ (1/2) (byt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0⊢ 0 < 1 / 2negt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0hL:¬Summable fun n ↦ 1 / (t ^ ↑n - 1)⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n norm_numAll goals completed! 🐙negt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0hL:¬Summable fun n ↦ 1 / (t ^ ↑n - 1)⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n)
      convert Set.infinite_univ (α := ℕ+)t:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0⊢ {n | 1 / 2 ≤ |1 / (t ^ ↑n - 1)|} = Set.univnegt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0hL:¬Summable fun n ↦ 1 / (t ^ ↑n - 1)⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n
      ext nt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0n:ℕ+⊢ n ∈ {n | 1 / 2 ≤ |1 / (t ^ ↑n - 1)|} ↔ n ∈ Set.univnegt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0hL:¬Summable fun n ↦ 1 / (t ^ ↑n - 1)⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n
      simp only [Set.mem_ofPred_eq, Set.mem_univ, iff_true]t:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0n:ℕ+⊢ 1 / 2 ≤ |1 / (t ^ ↑n - 1)|negt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0hL:¬Summable fun n ↦ 1 / (t ^ ↑n - 1)⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n
      have hden_bound : |t ^ (n : ℕ) - 1| ≤ 2 := byt:ℝht:|t| ≤ 1⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑nt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0n:ℕ+hden_bound:|t ^ ↑n - 1| ≤ 2⊢ 1 / 2 ≤ |1 / (t ^ ↑n - 1)|negt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0hL:¬Summable fun n ↦ 1 / (t ^ ↑n - 1)⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n
        calc |t ^ (n : ℕ) - 1| ≤ |t ^ (n : ℕ)| + |(1 : ℝ)| := abs_sub _ _
          _ ≤ 1 + 1 := byt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0n:ℕ+⊢ |t ^ ↑n| + |1| ≤ 1 + 1t:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0n:ℕ+hden_bound:|t ^ ↑n - 1| ≤ 2⊢ 1 / 2 ≤ |1 / (t ^ ↑n - 1)|negt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0hL:¬Summable fun n ↦ 1 / (t ^ ↑n - 1)⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n have := hbound nt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0n:ℕ+this:|t ^ ↑n| ≤ 1⊢ |t ^ ↑n| + |1| ≤ 1 + 1t:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0n:ℕ+hden_bound:|t ^ ↑n - 1| ≤ 2⊢ 1 / 2 ≤ |1 / (t ^ ↑n - 1)|negt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0hL:¬Summable fun n ↦ 1 / (t ^ ↑n - 1)⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n; rw [abs_onet:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0n:ℕ+this:|t ^ ↑n| ≤ 1⊢ |t ^ ↑n| + 1 ≤ 1 + 1t:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0n:ℕ+this:|t ^ ↑n| ≤ 1⊢ |t ^ ↑n| + 1 ≤ 1 + 1t:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0n:ℕ+hden_bound:|t ^ ↑n - 1| ≤ 2⊢ 1 / 2 ≤ |1 / (t ^ ↑n - 1)|negt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0hL:¬Summable fun n ↦ 1 / (t ^ ↑n - 1)⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n]t:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0n:ℕ+this:|t ^ ↑n| ≤ 1⊢ |t ^ ↑n| + 1 ≤ 1 + 1t:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0n:ℕ+hden_bound:|t ^ ↑n - 1| ≤ 2⊢ 1 / 2 ≤ |1 / (t ^ ↑n - 1)|negt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0hL:¬Summable fun n ↦ 1 / (t ^ ↑n - 1)⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n; linarithAll goals completed! 🐙t:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0n:ℕ+hden_bound:|t ^ ↑n - 1| ≤ 2⊢ 1 / 2 ≤ |1 / (t ^ ↑n - 1)|negt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0hL:¬Summable fun n ↦ 1 / (t ^ ↑n - 1)⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n
          _ = 2 := byt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0n:ℕ+⊢ 1 + 1 = 2t:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0n:ℕ+hden_bound:|t ^ ↑n - 1| ≤ 2⊢ 1 / 2 ≤ |1 / (t ^ ↑n - 1)|negt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0hL:¬Summable fun n ↦ 1 / (t ^ ↑n - 1)⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n norm_numt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0n:ℕ+hden_bound:|t ^ ↑n - 1| ≤ 2⊢ 1 / 2 ≤ |1 / (t ^ ↑n - 1)|negt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0hL:¬Summable fun n ↦ 1 / (t ^ ↑n - 1)⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑nt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0n:ℕ+hden_bound:|t ^ ↑n - 1| ≤ 2⊢ 1 / 2 ≤ |1 / (t ^ ↑n - 1)|negt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0hL:¬Summable fun n ↦ 1 / (t ^ ↑n - 1)⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n
      have hden_pos : 0 < |t ^ (n : ℕ) - 1| := byt:ℝht:|t| ≤ 1⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑nt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0n:ℕ+hden_bound:|t ^ ↑n - 1| ≤ 2hden_pos:0 < |t ^ ↑n - 1|⊢ 1 / 2 ≤ |1 / (t ^ ↑n - 1)|negt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0hL:¬Summable fun n ↦ 1 / (t ^ ↑n - 1)⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n
        rw [abs_pos,t:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0n:ℕ+hden_bound:|t ^ ↑n - 1| ≤ 2⊢ t ^ ↑n - 1 ≠ 0t:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0n:ℕ+hden_bound:|t ^ ↑n - 1| ≤ 2⊢ t ^ ↑n ≠ 1t:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0n:ℕ+hden_bound:|t ^ ↑n - 1| ≤ 2hden_pos:0 < |t ^ ↑n - 1|⊢ 1 / 2 ≤ |1 / (t ^ ↑n - 1)|negt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0hL:¬Summable fun n ↦ 1 / (t ^ ↑n - 1)⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n sub_ne_zerot:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0n:ℕ+hden_bound:|t ^ ↑n - 1| ≤ 2⊢ t ^ ↑n ≠ 1t:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0n:ℕ+hden_bound:|t ^ ↑n - 1| ≤ 2⊢ t ^ ↑n ≠ 1t:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0n:ℕ+hden_bound:|t ^ ↑n - 1| ≤ 2hden_pos:0 < |t ^ ↑n - 1|⊢ 1 / 2 ≤ |1 / (t ^ ↑n - 1)|negt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0hL:¬Summable fun n ↦ 1 / (t ^ ↑n - 1)⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n]t:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0n:ℕ+hden_bound:|t ^ ↑n - 1| ≤ 2⊢ t ^ ↑n ≠ 1t:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0n:ℕ+hden_bound:|t ^ ↑n - 1| ≤ 2hden_pos:0 < |t ^ ↑n - 1|⊢ 1 / 2 ≤ |1 / (t ^ ↑n - 1)|negt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0hL:¬Summable fun n ↦ 1 / (t ^ ↑n - 1)⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n; exact htn_ne_one nt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0n:ℕ+hden_bound:|t ^ ↑n - 1| ≤ 2hden_pos:0 < |t ^ ↑n - 1|⊢ 1 / 2 ≤ |1 / (t ^ ↑n - 1)|negt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0hL:¬Summable fun n ↦ 1 / (t ^ ↑n - 1)⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑nt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0n:ℕ+hden_bound:|t ^ ↑n - 1| ≤ 2hden_pos:0 < |t ^ ↑n - 1|⊢ 1 / 2 ≤ |1 / (t ^ ↑n - 1)|negt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0hL:¬Summable fun n ↦ 1 / (t ^ ↑n - 1)⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n
      rw [abs_div,t:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0n:ℕ+hden_bound:|t ^ ↑n - 1| ≤ 2hden_pos:0 < |t ^ ↑n - 1|⊢ 1 / 2 ≤ |1| / |t ^ ↑n - 1|t:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0n:ℕ+hden_bound:|t ^ ↑n - 1| ≤ 2hden_pos:0 < |t ^ ↑n - 1|⊢ 1 / 2 * |t ^ ↑n - 1| ≤ 1negt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0hL:¬Summable fun n ↦ 1 / (t ^ ↑n - 1)⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n abs_one,t:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0n:ℕ+hden_bound:|t ^ ↑n - 1| ≤ 2hden_pos:0 < |t ^ ↑n - 1|⊢ 1 / 2 ≤ 1 / |t ^ ↑n - 1|t:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0n:ℕ+hden_bound:|t ^ ↑n - 1| ≤ 2hden_pos:0 < |t ^ ↑n - 1|⊢ 1 / 2 * |t ^ ↑n - 1| ≤ 1negt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0hL:¬Summable fun n ↦ 1 / (t ^ ↑n - 1)⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n le_div_iff₀ hden_post:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0n:ℕ+hden_bound:|t ^ ↑n - 1| ≤ 2hden_pos:0 < |t ^ ↑n - 1|⊢ 1 / 2 * |t ^ ↑n - 1| ≤ 1t:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0n:ℕ+hden_bound:|t ^ ↑n - 1| ≤ 2hden_pos:0 < |t ^ ↑n - 1|⊢ 1 / 2 * |t ^ ↑n - 1| ≤ 1negt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0hL:¬Summable fun n ↦ 1 / (t ^ ↑n - 1)⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n]t:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0n:ℕ+hden_bound:|t ^ ↑n - 1| ≤ 2hden_pos:0 < |t ^ ↑n - 1|⊢ 1 / 2 * |t ^ ↑n - 1| ≤ 1negt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0hL:¬Summable fun n ↦ 1 / (t ^ ↑n - 1)⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n; linarithnegt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0hL:¬Summable fun n ↦ 1 / (t ^ ↑n - 1)⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑nnegt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0hL:¬Summable fun n ↦ 1 / (t ^ ↑n - 1)⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n
    -- RHS: `|τ(n) / t^n| = τ(n) / |t^n| ≥ τ(n) ≥ 1` since `|t^n| ≤ 1`.
    have hR : ¬ Summable (fun n : ℕ+ => ((n : ℕ).divisors.card : ℝ) / (t ^ (n : ℕ))) := by
      apply key _ 1 (byt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0hL:¬Summable fun n ↦ 1 / (t ^ ↑n - 1)⊢ 0 < 1negt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0hL:¬Summable fun n ↦ 1 / (t ^ ↑n - 1)hR:¬Summable fun n ↦ ↑(↑n).divisors.card / t ^ ↑n⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n norm_numAll goals completed! 🐙negt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0hL:¬Summable fun n ↦ 1 / (t ^ ↑n - 1)hR:¬Summable fun n ↦ ↑(↑n).divisors.card / t ^ ↑n⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n)
      convert Set.infinite_univ (α := ℕ+)t:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0hL:¬Summable fun n ↦ 1 / (t ^ ↑n - 1)⊢ {n | 1 ≤ |↑(↑n).divisors.card / t ^ ↑n|} = Set.univnegt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0hL:¬Summable fun n ↦ 1 / (t ^ ↑n - 1)hR:¬Summable fun n ↦ ↑(↑n).divisors.card / t ^ ↑n⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n
      ext nt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0hL:¬Summable fun n ↦ 1 / (t ^ ↑n - 1)n:ℕ+⊢ n ∈ {n | 1 ≤ |↑(↑n).divisors.card / t ^ ↑n|} ↔ n ∈ Set.univnegt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0hL:¬Summable fun n ↦ 1 / (t ^ ↑n - 1)hR:¬Summable fun n ↦ ↑(↑n).divisors.card / t ^ ↑n⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n
      simp only [Set.mem_ofPred_eq, Set.mem_univ, iff_true]t:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0hL:¬Summable fun n ↦ 1 / (t ^ ↑n - 1)n:ℕ+⊢ 1 ≤ |↑(↑n).divisors.card / t ^ ↑n|negt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0hL:¬Summable fun n ↦ 1 / (t ^ ↑n - 1)hR:¬Summable fun n ↦ ↑(↑n).divisors.card / t ^ ↑n⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n
      rw [abs_div,t:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0hL:¬Summable fun n ↦ 1 / (t ^ ↑n - 1)n:ℕ+⊢ 1 ≤ |↑(↑n).divisors.card| / |t ^ ↑n|t:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0hL:¬Summable fun n ↦ 1 / (t ^ ↑n - 1)n:ℕ+⊢ 1 * |t ^ ↑n| ≤ |↑(↑n).divisors.card|negt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0hL:¬Summable fun n ↦ 1 / (t ^ ↑n - 1)hR:¬Summable fun n ↦ ↑(↑n).divisors.card / t ^ ↑n⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n le_div_iff₀ (abs_pos.mpr (htn_ne_zero n))t:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0hL:¬Summable fun n ↦ 1 / (t ^ ↑n - 1)n:ℕ+⊢ 1 * |t ^ ↑n| ≤ |↑(↑n).divisors.card|t:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0hL:¬Summable fun n ↦ 1 / (t ^ ↑n - 1)n:ℕ+⊢ 1 * |t ^ ↑n| ≤ |↑(↑n).divisors.card|negt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0hL:¬Summable fun n ↦ 1 / (t ^ ↑n - 1)hR:¬Summable fun n ↦ ↑(↑n).divisors.card / t ^ ↑n⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n]t:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0hL:¬Summable fun n ↦ 1 / (t ^ ↑n - 1)n:ℕ+⊢ 1 * |t ^ ↑n| ≤ |↑(↑n).divisors.card|negt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0hL:¬Summable fun n ↦ 1 / (t ^ ↑n - 1)hR:¬Summable fun n ↦ ↑(↑n).divisors.card / t ^ ↑n⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n
      rw [abs_of_nonneg (byt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0hL:¬Summable fun n ↦ 1 / (t ^ ↑n - 1)n:ℕ+⊢ 0 ≤ ↑(↑n).divisors.cardt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0hL:¬Summable fun n ↦ 1 / (t ^ ↑n - 1)n:ℕ+⊢ 1 * |t ^ ↑n| ≤ ↑(↑n).divisors.cardnegt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0hL:¬Summable fun n ↦ 1 / (t ^ ↑n - 1)hR:¬Summable fun n ↦ ↑(↑n).divisors.card / t ^ ↑n⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n positivityAll goals completed! 🐙t:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0hL:¬Summable fun n ↦ 1 / (t ^ ↑n - 1)n:ℕ+⊢ 1 * |t ^ ↑n| ≤ ↑(↑n).divisors.cardnegt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0hL:¬Summable fun n ↦ 1 / (t ^ ↑n - 1)hR:¬Summable fun n ↦ ↑(↑n).divisors.card / t ^ ↑n⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n : (0:ℝ) ≤ ((n : ℕ).divisors.card : ℝ))]t:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0hL:¬Summable fun n ↦ 1 / (t ^ ↑n - 1)n:ℕ+⊢ 1 * |t ^ ↑n| ≤ ↑(↑n).divisors.cardnegt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0hL:¬Summable fun n ↦ 1 / (t ^ ↑n - 1)hR:¬Summable fun n ↦ ↑(↑n).divisors.card / t ^ ↑n⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n
      have := hbound nt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0hL:¬Summable fun n ↦ 1 / (t ^ ↑n - 1)n:ℕ+this:|t ^ ↑n| ≤ 1⊢ 1 * |t ^ ↑n| ≤ ↑(↑n).divisors.cardnegt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0hL:¬Summable fun n ↦ 1 / (t ^ ↑n - 1)hR:¬Summable fun n ↦ ↑(↑n).divisors.card / t ^ ↑n⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n
      have := hcard_pos nt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0hL:¬Summable fun n ↦ 1 / (t ^ ↑n - 1)n:ℕ+this✝:|t ^ ↑n| ≤ 1this:1 ≤ ↑(↑n).divisors.card⊢ 1 * |t ^ ↑n| ≤ ↑(↑n).divisors.cardnegt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0hL:¬Summable fun n ↦ 1 / (t ^ ↑n - 1)hR:¬Summable fun n ↦ ↑(↑n).divisors.card / t ^ ↑n⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n
      nlinarithnegt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0hL:¬Summable fun n ↦ 1 / (t ^ ↑n - 1)hR:¬Summable fun n ↦ ↑(↑n).divisors.card / t ^ ↑n⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑nnegt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0hL:¬Summable fun n ↦ 1 / (t ^ ↑n - 1)hR:¬Summable fun n ↦ ↑(↑n).divisors.card / t ^ ↑n⊢ ∑' (n : ℕ+), 1 / (t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑n
    rw [tsum_eq_zero_of_not_summable hL,negt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0hL:¬Summable fun n ↦ 1 / (t ^ ↑n - 1)hR:¬Summable fun n ↦ ↑(↑n).divisors.card / t ^ ↑n⊢ 0 = ∑' (n : ℕ+), ↑(↑n).divisors.card / t ^ ↑nAll goals completed! 🐙 tsum_eq_zero_of_not_summable hRnegt:ℝht:|t| ≤ 1key:∀ (f : ℕ+ → ℝ) (c : ℝ), 0 < c → {n | c ≤ |f n|}.Infinite → ¬Summable fhcard_pos:∀ (n : ℕ+), 1 ≤ ↑(↑n).divisors.cardht1:¬t = 1ht0:¬t = 0htneg1:¬t = -1habs_lt:|t| < 1habs_pos:0 < |t|hbound:∀ (n : ℕ+), |t ^ ↑n| ≤ 1htn_ne_one:∀ (n : ℕ+), t ^ ↑n ≠ 1htn_ne_zero:∀ (n : ℕ+), t ^ ↑n ≠ 0hL:¬Summable fun n ↦ 1 / (t ^ ↑n - 1)hR:¬Summable fun n ↦ ↑(↑n).divisors.card / t ^ ↑n⊢ 0 = 0All goals completed! 🐙]All goals completed! 🐙

The classical Lambert series identity: $\sum_{n=1}^\infty \frac{1}{t^n - 1} = \sum_{n=1}^\infty \frac{\tau(n)}{t^n}$, where $\tau(n)$ counts the divisors of $n$.

@[category textbook, AMS 11]
theorem lambert_series_eq_num_divisor_sum : ∀ t : ℚ,
     ∑' n : ℕ+, 1 / ((t : ℝ) ^ (n : ℕ) - 1) =
     ∑' n : ℕ+, (n : ℕ).divisors.card / ((t : ℝ) ^ (n : ℕ)) := by⊢ ∀ (t : ℚ), ∑' (n : ℕ+), 1 / (↑t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / ↑t ^ ↑n
  intro tt:ℚ⊢ ∑' (n : ℕ+), 1 / (↑t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / ↑t ^ ↑n
  obtain ht | ht := lt_or_ge 1 |(t : ℝ)|inlt:ℚht:1 < |↑t|⊢ ∑' (n : ℕ+), 1 / (↑t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / ↑t ^ ↑ninrt:ℚht:|↑t| ≤ 1⊢ ∑' (n : ℕ+), 1 / (↑t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / ↑t ^ ↑n
  ·inlt:ℚht:1 < |↑t|⊢ ∑' (n : ℕ+), 1 / (↑t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / ↑t ^ ↑n exact lambert_convergent (t : ℝ) htAll goals completed! 🐙
  ·inrt:ℚht:|↑t| ≤ 1⊢ ∑' (n : ℕ+), 1 / (↑t ^ ↑n - 1) = ∑' (n : ℕ+), ↑(↑n).divisors.card / ↑t ^ ↑n exact lambert_divergent (t : ℝ) htAll goals completed! 🐙

end Erdos1049