A256358 Decimal expansion of log(sqrt(Pi/2)).
2, 2, 5, 7, 9, 1, 3, 5, 2, 6, 4, 4, 7, 2, 7, 4, 3, 2, 3, 6, 3, 0, 9, 7, 6, 1, 4, 9, 4, 7, 4, 4, 1, 0, 7, 1, 7, 8, 5, 8, 9, 7, 3, 3, 9, 2, 7, 7, 5, 2, 8, 1, 5, 8, 6, 9, 6, 4, 7, 1, 5, 3, 0, 9, 8, 9, 3, 7, 2, 0, 7, 3, 9, 5, 7, 5, 6, 5, 6, 8, 2, 0, 8, 8, 8, 7, 9, 9, 7, 1, 6, 3, 9, 5, 3, 5, 5, 1, 0, 0, 8, 0, 0, 0, 4
Offset: 0
Examples
0.22579135264472743236309761494744107178589733927752815869647153...
Links
- G. C. Greubel, Table of n, a(n) for n = 0..5000
- J.-F. Alcover, Plot of harmonic sum G(x) for x >= 0 .
- Henri Cohen, Fernando Rodriguez Villegas, Don Zagier, Convergence Acceleration of Alternating Series, Exp. Math. 9 (1) (2000) 3-12.
- Costas Efthimiou, Problem 1838, Mathematics Magazine, Vol. 83, No. 1 (2010), p. 65; A weakly convergent series of logs, Solution to Problem 1838 by Tiberiu Trif, ibid., Vol. 84, No. 1 (2011), pp. 65-67.
- Philippe Flajolet, Xavier Gourdon, and Philippe Dumas, Mellin Transforms and Asymptotics: Harmonic Sums.
- Eric Weisstein's World of Mathematics, Dirichlet eta function.
Programs
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Mathematica
RealDigits[Log[Sqrt[Pi/2]], 10, 105] // First RealDigits[DirichletEta'[0], 10, 110][[1]] (* Eric W. Weisstein, Jan 06 2024 *)
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PARI
log(sqrt(Pi/2)) \\ G. C. Greubel, Jan 09 2017
Formula
Given the harmonic sum G(x) = Sum_{k>=1} (-1)^k*log(k)*exp(-k^2*x), lim_{x->0} G(x) = log(sqrt(Pi/2)).
Integral_{x=0..oo} G(x) dx = (Pi^2/12)*log(2) + zeta'(2)/2 = (Pi^2/12)*(EulerGamma + log(4*Pi) - 12*log(Glaisher)) = 0.1013165781635...
G'(0) = 7*zeta'(-2) = -7*zeta(3)/(4*Pi^2) = -0.2131391994...
Equals Integral_{-oo..+oo} -log(1/2 + i*z)/(exp(-Pi*z) + exp(Pi*z)) dz, where i is the imaginary unit. - Peter Luschny, Apr 08 2018
Equals Sum_{n>=0} Sum_{m>=1} (-1)^(m+n) * log(m+n)/(m+n) (Efthimiou, 2010). - Amiram Eldar, Apr 09 2022
Equals A094642/2. - R. J. Mathar, Jun 15 2023
Comments