A265162 Decimal expansion of Sum_{k>=1} (-1)^k*log(k)/sqrt(k).

1, 9, 3, 2, 8, 8, 8, 3, 1, 6, 3, 9, 2, 8, 2, 7, 3, 8, 9, 6, 4, 6, 1, 5, 4, 5, 9, 3, 5, 5, 2, 3, 8, 1, 1, 4, 2, 9, 5, 2, 7, 0, 2, 2, 2, 5, 2, 9, 2, 2, 1, 9, 9, 2, 2, 9, 3, 6, 0, 4, 8, 1, 0, 3, 3, 4, 4, 0, 1, 6, 6, 6, 4, 4, 4, 4, 6, 8, 9, 8, 7, 3, 4, 9, 8, 6, 8, 0, 9, 2, 0, 8, 7, 7, 7, 8, 1, 6, 3, 6, 8, 4, 5, 7, 2, 6

Offset: 0

Author

Vaclav Kotesovec, Dec 03 2015

Keywords

• nonn
• cons

Comments

Differentiation of Sum_{k>=1} (-1)^k/k^s = -(2^s-2)*zeta(s)/2^s with respect to s gives -Sum_{k>=1} (-1)^k*log(k)/k^s = -2^(1-s)*log(2)*zeta(s) - (1-2^(1-s))*zeta'(s), where zeta(.) and zeta'(.) are the Riemann zeta function and its derivative. - R. J. Mathar, Apr 17 2019, typo in the first formula corrected by Vaclav Kotesovec, Jan 11 2024

Examples

			0.1932888316392827389646154593552381142952702225292219922936048103344...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..10000
• Henri Cohen, Fernando Rodriguez Villegas, and Don Zagier, Convergence Acceleration of Alternating Series, Exp. Math. 9 (1) (2000) 3-12.
• Eric Weisstein's World of Mathematics, Dirichlet Eta Function.

Crossrefs

Cf. A091812, A113024.

Programs

Maple

evalf(sum((-1)^k*log(k)/sqrt(k), k=1..infinity), 120);

Mathematica

RealDigits[((3-Sqrt[2])*Log[2]/2 - (Sqrt[2]-1)*(2*EulerGamma + Pi + 2*Log[Pi])/4) * Zeta[1/2], 10, 106][[1]]
RealDigits[DirichletEta'[1/2], 10, 110][[1]] (* Eric W. Weisstein, Jan 08 2024 *)

PARI

((3-sqrt(2))*log(2)/2 - (sqrt(2)-1)*(2*Euler + Pi + 2*log(Pi))/4)* zeta(1/2) \\ G. C. Greubel, Apr 15 2018

Formula

Equals ((3-sqrt(2))*log(2)/2 - (sqrt(2)-1)*(2*gamma + Pi + 2*log(Pi))/4) * zeta(1/2), where gamma is the Euler-Mascheroni constant A001620.

A265162/A113024 = gamma/2 + Pi/4 - (1/2 + sqrt(2))*log(2) + log(Pi)/2.