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Level A · Machine-checkable Hard Analysis P-flint-cookson-hills

Convergence of the Flint Hills and Cookson Hills series

The Flint Hills series summing csc(n)^2 / n^3 from n=1 to ∞ converges. (Note that we 0-index the series below.)

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim.

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Cite 
@misc{cairn-flint-cookson-hills,
  title        = {Convergence of the Flint Hills and Cookson Hills series},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/flint-cookson-hills}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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The problem

The question

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.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Wikipedia.FlintCooksonHills (2 statements). answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem flint_hills_series_converges :
    answer(sorry) ↔
      Summable (fun n : ℕ =>
        1 / ((((n + 1) : ℝ)^3) * (Real.sin (n + 1)^2)))
theorem cookson_hills_series_converges :
    answer(sorry) ↔
      Summable (fun n : ℕ =>
        1 / ((((n + 1) : ℝ)^3) * (Real.cos (n + 1)^2)))

What counts as progress

  • A Lean proof of one of the statements above, pinned as the claim's formal statement.
  • Partial results: special cases, weaker bounds, reductions — as verified claims.
  • Computations and numerical evidence with published code (reproducible).
  • Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

Source and licence

Imported from Formal Conjectures (Wikipedia), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.