A352527 Decimal expansion of Sum_(k>=1) (-1)^k * zeta(2k)/(2k) (negated).
6, 5, 0, 9, 2, 3, 1, 9, 9, 3, 0, 1, 8, 5, 6, 3, 3, 8, 8, 8, 5, 2, 1, 6, 8, 3, 1, 5, 0, 3, 9, 4, 7, 6, 6, 5, 0, 6, 5, 5, 0, 8, 7, 5, 7, 1, 3, 9, 7, 2, 2, 5, 9, 1, 9, 9, 8, 3, 8, 2, 4, 8, 2, 1, 0, 6, 4, 0, 7, 4, 3, 1, 1, 3, 0, 4, 9, 6, 7, 0, 7, 0, 6, 4, 5, 5, 8, 5, 9, 5, 0, 9, 4, 0, 9, 4
Offset: 1
Examples
-0.65092319930185633888521683150394766...
Links
- Les-mathematiques.net, Somme arctan(1/k).
- Cornel Ioan Vălean, Problem 11924, The American Mathematical Monthly, Vol. 123, No. 7 (2016), p. 722; An Integral with Fractional Part of Tangent, Solution to Problem 11924 by Edward and Roberta White, ibid., Vol. 125, No. 5 (2018), pp. 468-470.
Programs
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Maple
evalf(log(Pi/sinh(Pi)) / 2, 100);
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Mathematica
RealDigits[Log[Pi/Sinh[Pi]]/2, 10, 100][[1]] (* Amiram Eldar, Mar 19 2022 *)
Formula
Equals log(Pi/sinh(Pi)) / 2.
Equals Integral_{x=0..Pi/2} ({tan(x)}/tan(x) - 1) dx, where {x} = x - floor(x) is the fractional part of x (Vălean, 2016). - Amiram Eldar, Feb 08 2024