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import FormalConjecturesUtilConvergence of the Borwein Series with Sinusoidal Coefficient
Borwein, J.; Bailey, D.; Girgensohn, R.
namespace BorweinSineSeries
Does the series $$ \sum_{n=1}^{\infty} \frac{\left(\frac{2}{3} + \frac{1}{3}\sin n\right)^n}{n} $$ converge?
After computing approximately $10^7$ terms, the partial sums approximate $2.163$.
See https://arxiv.org/abs/2007.11017 for a proof of the convergence, relying on an irrationality measure for pi.
Also see https://github.com/AxiomMath/gdm-formal-conjectures/blob/main/docs/BorweinSineSeries.md for a partial formalization of the conjecture, conditional on such an irrationality measure of pi (cf https://arxiv.org/abs/1912.06345).
@[category research solved, formal_proof using lean4 at "https://github.com/AxiomMath/gdm-formal-conjectures/blob/main/BorweinSineSeries/solution.lean", AMS 26 40]
theorem borwein_sine_series :
answer(sorry) ↔
Summable fun n : ℕ+ ↦ ((2 / 3 + 1 / 3 * Real.sin (n : ℝ)) ^ (n : ℕ)) / (n : ℝ) := ⊢ True ↔ Summable fun n => (2 / 3 + 1 / 3 * Real.sin ↑↑n) ^ ↑n / ↑↑n
All goals completed! 🐙
end BorweinSineSeries