Theorems · Theorem · real analysis
Real.tendsto_sum_pi_div_four
- #26 of the 100 theorems: Leibniz’s Series for Pi
Filter.Tendsto (fun k => ∑ i ∈ Finset.range k, (-1) ^ i / (2 * ↑i + 1)) Filter.atTop (nhds (Real.pi / 4))
Leibniz's series for `π`. The alternating sum of odd number reciprocals is π / 4,
proved by using Abel's limit theorem to extend the Maclaurin series of arctan to 1.
- Defined in
- Mathlib.Analysis.Real.Pi.Leibniz
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 277 from the axioms · uses propext, Classical.choice, Quot.sound
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Cites67
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setproof · cited by 53,352
- Realstatement and proof · cited by 25,697
- nhdsstatement and proof · cited by 5,554
- Norm.normproof · cited by 5,413
- Finset.sumstatement and proof · cited by 5,195
- mul_oneproof · cited by 3,885
- Filter.Tendstostatement and proof · cited by 3,814
- Nat.cast_oneproof · cited by 2,501
- Filter.atTopstatement and proof · cited by 2,405
- SummationFilter.unconditionalproof · cited by 2,068
- le_reflproof · cited by 2,061
- nhdsWithinproof · cited by 1,912
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