Convergence of the Flint Hills and Cookson Hills series
Convergence of the Flint Hills and Cookson Hills series
Problem statement
flint_hills_series_converges
The Flint Hills series summing from to converges. (Note that we 0-index the series below.)
cookson_hills_series_converges
The Cookson Hills series summing from to converges.
Lean 4
Formal statement
Lean 4/-
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-/
import FormalConjecturesUtil
/-!
# Convergence of the Flint Hills and Cookson Hills series
*References:*
- [Wikipedia: Examples of numerical series](https://en.wikipedia.org/wiki/Series_(mathematics)#Examples_of_numerical_series)
- [MathWorld: Flint Hills Series](https://mathworld.wolfram.com/FlintHillsSeries.html)
- [Alekseyev, On the Flint Hills series](https://doi.org/10.48550/arXiv.1104.5100)
- [MathWorld: Cookson Hills Series](https://mathworld.wolfram.com/CooksonHillsSeries.html)
-/
namespace FlintCooksonHills
/--
The Flint Hills series summing $csc(n)^2 / n^3$ from $n=1$ to $\infty$ converges.
(Note that we 0-index the series below.)
-/
@[category research open, AMS 40]
theorem flint_hills_series_converges :
answer(sorry) ↔
Summable (fun n : ℕ =>
1 / ((((n + 1) : ℝ)^3) * (Real.sin (n + 1)^2))) := by
sorry
/--
The Cookson Hills series summing $sec(n)^2 / n^3$ from $n=1$ to $\infty$ converges.
-/
@[category research open, AMS 40]
theorem cookson_hills_series_converges :
answer(sorry) ↔
Summable (fun n : ℕ =>
1 / ((((n + 1) : ℝ)^3) * (Real.cos (n + 1)^2))) := by
sorry
end FlintCooksonHills
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Background
Source snapshot inspected on 10 September 2026. Subsequent research status and the accuracy of source formalizations have not been independently certified.
Fixed commit b82b08faa9006484021c12005ab41287fb2ffb69. This file contains 2 source-labeled research-open declarations. Other declarations in the attached file can be definitions, solved results or tests. The file is preserved as source material; it has not been compiled or proved by this import.
Copyright The Formal Conjectures Authors. Code: Apache-2.0; other original material: CC BY 4.0. Third-party sources retain their own licenses. Original source files and notices preserved.
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