FormalConjectures.ErdosProblems.«68»

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import FormalConjecturesUtil

Erdős Problem 68

Reference: erdosproblems.com/68

namespace Erdos68

Is $$\sum_{n=2}^\infty \frac{1}{n!-1}$$ irrational?

@[category research open, AMS 11]
theorem erdos_68 :
    answer(sorry) ↔ Irrational (∑' n : ℕ, 1 / ((n + 2).factorial - 1 : ℝ)) := by⊢ True ↔ Irrational (∑' (n : ℕ), 1 / (↑(n + 2).factorial - 1))
  sorryAll goals completed! 🐙

$$\sum_{n=2}^\infty \frac{1}{n!-1} = \sum_{n=2}^\infty \sum_{k=1}^\infty \frac{1}{(n!)^k}$$


@[category textbook, AMS 11]
theorem sum_factorial_inv_eq_geometric :
    let f (n k : ℕ) : ℝ := 1 / ((n + 2).factorial : ℝ) ^ (k + 1)
    ∑' n : ℕ, (1 : ℝ) / ((n + 2).factorial - 1) = ∑' n : ℕ, ∑' k : ℕ, f n k := by⊢ let f := fun n k ↦ 1 / ↑(n + 2).factorial ^ (k + 1);
∑' (n : ℕ), 1 / (↑(n + 2).factorial - 1) = ∑' (n : ℕ) (k : ℕ), f n k
  intro ff:ℕ → ℕ → ℝ := fun n k ↦ 1 / ↑(n + 2).factorial ^ (k + 1)⊢ ∑' (n : ℕ), 1 / (↑(n + 2).factorial - 1) = ∑' (n : ℕ) (k : ℕ), f n k
  apply tsum_congrf:ℕ → ℕ → ℝ := fun n k ↦ 1 / ↑(n + 2).factorial ^ (k + 1)⊢ ∀ (b : ℕ), 1 / (↑(b + 2).factorial - 1) = ∑' (k : ℕ), f b k
  intro nf:ℕ → ℕ → ℝ := fun n k ↦ 1 / ↑(n + 2).factorial ^ (k + 1)n:ℕ⊢ 1 / (↑(n + 2).factorial - 1) = ∑' (k : ℕ), f n k
  symmf:ℕ → ℕ → ℝ := fun n k ↦ 1 / ↑(n + 2).factorial ^ (k + 1)n:ℕ⊢ ∑' (k : ℕ), f n k = 1 / (↑(n + 2).factorial - 1)
  -- The inner sum is a geometric series with ratio r = ((n + 2)!)⁻¹.
  set r : ℝ := ((n + 2).factorial : ℝ)⁻¹ with hr_deff:ℕ → ℕ → ℝ := fun n k ↦ 1 / ↑(n + 2).factorial ^ (k + 1)n:ℕr:ℝ := (↑(n + 2).factorial)⁻¹hr_def:r = (↑(n + 2).factorial)⁻¹⊢ ∑' (k : ℕ), f n k = 1 / (↑(n + 2).factorial - 1)
  have hr_nonneg : 0 ≤ r := by⊢ let f := fun n k ↦ 1 / ↑(n + 2).factorial ^ (k + 1);
∑' (n : ℕ), 1 / (↑(n + 2).factorial - 1) = ∑' (n : ℕ) (k : ℕ), f n kf:ℕ → ℕ → ℝ := fun n k ↦ 1 / ↑(n + 2).factorial ^ (k + 1)n:ℕr:ℝ := (↑(n + 2).factorial)⁻¹hr_def:r = (↑(n + 2).factorial)⁻¹hr_nonneg:0 ≤ r⊢ ∑' (k : ℕ), f n k = 1 / (↑(n + 2).factorial - 1) positivityf:ℕ → ℕ → ℝ := fun n k ↦ 1 / ↑(n + 2).factorial ^ (k + 1)n:ℕr:ℝ := (↑(n + 2).factorial)⁻¹hr_def:r = (↑(n + 2).factorial)⁻¹hr_nonneg:0 ≤ r⊢ ∑' (k : ℕ), f n k = 1 / (↑(n + 2).factorial - 1)f:ℕ → ℕ → ℝ := fun n k ↦ 1 / ↑(n + 2).factorial ^ (k + 1)n:ℕr:ℝ := (↑(n + 2).factorial)⁻¹hr_def:r = (↑(n + 2).factorial)⁻¹hr_nonneg:0 ≤ r⊢ ∑' (k : ℕ), f n k = 1 / (↑(n + 2).factorial - 1)
  have hr_lt_one : r < 1 := inv_lt_one_of_one_lt₀ (byf:ℕ → ℕ → ℝ := fun n k ↦ 1 / ↑(n + 2).factorial ^ (k + 1)n:ℕr:ℝ := (↑(n + 2).factorial)⁻¹hr_def:r = (↑(n + 2).factorial)⁻¹hr_nonneg:0 ≤ r⊢ 1 < ↑(n + 2).factorialf:ℕ → ℕ → ℝ := fun n k ↦ 1 / ↑(n + 2).factorial ^ (k + 1)n:ℕr:ℝ := (↑(n + 2).factorial)⁻¹hr_def:r = (↑(n + 2).factorial)⁻¹hr_nonneg:0 ≤ rhr_lt_one:r < 1⊢ ∑' (k : ℕ), f n k = 1 / (↑(n + 2).factorial - 1) simpAll goals completed! 🐙f:ℕ → ℕ → ℝ := fun n k ↦ 1 / ↑(n + 2).factorial ^ (k + 1)n:ℕr:ℝ := (↑(n + 2).factorial)⁻¹hr_def:r = (↑(n + 2).factorial)⁻¹hr_nonneg:0 ≤ rhr_lt_one:r < 1⊢ ∑' (k : ℕ), f n k = 1 / (↑(n + 2).factorial - 1))f:ℕ → ℕ → ℝ := fun n k ↦ 1 / ↑(n + 2).factorial ^ (k + 1)n:ℕr:ℝ := (↑(n + 2).factorial)⁻¹hr_def:r = (↑(n + 2).factorial)⁻¹hr_nonneg:0 ≤ rhr_lt_one:r < 1⊢ ∑' (k : ℕ), f n k = 1 / (↑(n + 2).factorial - 1)
  -- Geometric series: HasSum (fun k ↦ r ^ k) ((1 - r)⁻¹)
  have hgeom := hasSum_geometric_of_lt_one hr_nonneg hr_lt_onef:ℕ → ℕ → ℝ := fun n k ↦ 1 / ↑(n + 2).factorial ^ (k + 1)n:ℕr:ℝ := (↑(n + 2).factorial)⁻¹hr_def:r = (↑(n + 2).factorial)⁻¹hr_nonneg:0 ≤ rhr_lt_one:r < 1hgeom:HasSum (fun n ↦ r ^ n) (1 - r)⁻¹⊢ ∑' (k : ℕ), f n k = 1 / (↑(n + 2).factorial - 1)
  -- Multiply by r to shift the index: HasSum (fun k ↦ r * r ^ k) (r * (1 - r)⁻¹)
  have hshift := hgeom.mul_left rf:ℕ → ℕ → ℝ := fun n k ↦ 1 / ↑(n + 2).factorial ^ (k + 1)n:ℕr:ℝ := (↑(n + 2).factorial)⁻¹hr_def:r = (↑(n + 2).factorial)⁻¹hr_nonneg:0 ≤ rhr_lt_one:r < 1hgeom:HasSum (fun n ↦ r ^ n) (1 - r)⁻¹hshift:HasSum (fun i ↦ r * r ^ i) (r * (1 - r)⁻¹)⊢ ∑' (k : ℕ), f n k = 1 / (↑(n + 2).factorial - 1)
  -- Each summand satisfies f n k = r * r ^ k.
  have hf_eq : ∀ k, f n k = r * r ^ k := fun k => byf:ℕ → ℕ → ℝ := fun n k ↦ 1 / ↑(n + 2).factorial ^ (k + 1)n:ℕr:ℝ := (↑(n + 2).factorial)⁻¹hr_def:r = (↑(n + 2).factorial)⁻¹hr_nonneg:0 ≤ rhr_lt_one:r < 1hgeom:HasSum (fun n ↦ r ^ n) (1 - r)⁻¹hshift:HasSum (fun i ↦ r * r ^ i) (r * (1 - r)⁻¹)k:ℕ⊢ f n k = r * r ^ kf:ℕ → ℕ → ℝ := fun n k ↦ 1 / ↑(n + 2).factorial ^ (k + 1)n:ℕr:ℝ := (↑(n + 2).factorial)⁻¹hr_def:r = (↑(n + 2).factorial)⁻¹hr_nonneg:0 ≤ rhr_lt_one:r < 1hgeom:HasSum (fun n ↦ r ^ n) (1 - r)⁻¹hshift:HasSum (fun i ↦ r * r ^ i) (r * (1 - r)⁻¹)hf_eq:∀ (k : ℕ), f n k = r * r ^ k⊢ ∑' (k : ℕ), f n k = 1 / (↑(n + 2).factorial - 1) simp only [f, hr_def]f:ℕ → ℕ → ℝ := fun n k ↦ 1 / ↑(n + 2).factorial ^ (k + 1)n:ℕr:ℝ := (↑(n + 2).factorial)⁻¹hr_def:r = (↑(n + 2).factorial)⁻¹hr_nonneg:0 ≤ rhr_lt_one:r < 1hgeom:HasSum (fun n ↦ r ^ n) (1 - r)⁻¹hshift:HasSum (fun i ↦ r * r ^ i) (r * (1 - r)⁻¹)k:ℕ⊢ 1 / ↑(n + 2).factorial ^ (k + 1) = (↑(n + 2).factorial)⁻¹ * (↑(n + 2).factorial)⁻¹ ^ kf:ℕ → ℕ → ℝ := fun n k ↦ 1 / ↑(n + 2).factorial ^ (k + 1)n:ℕr:ℝ := (↑(n + 2).factorial)⁻¹hr_def:r = (↑(n + 2).factorial)⁻¹hr_nonneg:0 ≤ rhr_lt_one:r < 1hgeom:HasSum (fun n ↦ r ^ n) (1 - r)⁻¹hshift:HasSum (fun i ↦ r * r ^ i) (r * (1 - r)⁻¹)hf_eq:∀ (k : ℕ), f n k = r * r ^ k⊢ ∑' (k : ℕ), f n k = 1 / (↑(n + 2).factorial - 1); ringf:ℕ → ℕ → ℝ := fun n k ↦ 1 / ↑(n + 2).factorial ^ (k + 1)n:ℕr:ℝ := (↑(n + 2).factorial)⁻¹hr_def:r = (↑(n + 2).factorial)⁻¹hr_nonneg:0 ≤ rhr_lt_one:r < 1hgeom:HasSum (fun n ↦ r ^ n) (1 - r)⁻¹hshift:HasSum (fun i ↦ r * r ^ i) (r * (1 - r)⁻¹)hf_eq:∀ (k : ℕ), f n k = r * r ^ k⊢ ∑' (k : ℕ), f n k = 1 / (↑(n + 2).factorial - 1)f:ℕ → ℕ → ℝ := fun n k ↦ 1 / ↑(n + 2).factorial ^ (k + 1)n:ℕr:ℝ := (↑(n + 2).factorial)⁻¹hr_def:r = (↑(n + 2).factorial)⁻¹hr_nonneg:0 ≤ rhr_lt_one:r < 1hgeom:HasSum (fun n ↦ r ^ n) (1 - r)⁻¹hshift:HasSum (fun i ↦ r * r ^ i) (r * (1 - r)⁻¹)hf_eq:∀ (k : ℕ), f n k = r * r ^ k⊢ ∑' (k : ℕ), f n k = 1 / (↑(n + 2).factorial - 1)
  -- Evaluate ∑' k, f n k = r * (1 - r)⁻¹ = 1 / ((n + 2)! - 1).
  exact ((hshift.congr_fun hf_eq).tsum_eq.trans (byf:ℕ → ℕ → ℝ := fun n k ↦ 1 / ↑(n + 2).factorial ^ (k + 1)n:ℕr:ℝ := (↑(n + 2).factorial)⁻¹hr_def:r = (↑(n + 2).factorial)⁻¹hr_nonneg:0 ≤ rhr_lt_one:r < 1hgeom:HasSum (fun n ↦ r ^ n) (1 - r)⁻¹hshift:HasSum (fun i ↦ r * r ^ i) (r * (1 - r)⁻¹)hf_eq:∀ (k : ℕ), f n k = r * r ^ k⊢ r * (1 - r)⁻¹ = 1 / (↑(n + 2).factorial - 1) simp only [hr_def]f:ℕ → ℕ → ℝ := fun n k ↦ 1 / ↑(n + 2).factorial ^ (k + 1)n:ℕr:ℝ := (↑(n + 2).factorial)⁻¹hr_def:r = (↑(n + 2).factorial)⁻¹hr_nonneg:0 ≤ rhr_lt_one:r < 1hgeom:HasSum (fun n ↦ r ^ n) (1 - r)⁻¹hshift:HasSum (fun i ↦ r * r ^ i) (r * (1 - r)⁻¹)hf_eq:∀ (k : ℕ), f n k = r * r ^ k⊢ (↑(n + 2).factorial)⁻¹ * (1 - (↑(n + 2).factorial)⁻¹)⁻¹ = 1 / (↑(n + 2).factorial - 1); field_simpAll goals completed! 🐙))

end Erdos68