BorweinSineSeries.lean`

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

1 declarations, 1 contain sorry, 0 declared as axioms

Convergence of the Borwein Series with Sinusoidal Coefficient

References:

MathWorld, Harmonic Series
Borwein, J.; Bailey, D.; Girgensohn, R. Experimentation in Mathematics: Computational Paths to Discovery, A K Peters, 2004, p. 56.

Imports (1)

FormalConjecturesUtil

theorem BorweinSineSeries.borwein_sine_series Does the series $$ \sum_{n=1}^{\infty} \frac{\left(\frac{2}{3} + \frac{1}{3}\sin n\right)^n}{n} $$ converge? sorry line 51 source
Source documentation
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).

depends on sorry what this means
Statement uses: Iff, True, Summable, Real, PNat, HDiv.hDiv, HPow.hPow, Nat, HAdd.hAdd, OfNat.ofNat, HMul.hMul, Real.sin, SummationFilter.unconditional
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
48 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem borwein_sine_series :
    answer(True) ↔
      Summable fun n : ℕ+ ↦ ((2 / 3 + 1 / 3 * Real.sin (n : ℝ)) ^ (n : ℕ)) / (n : ℝ) := by
…
```