OFFSET
0,2
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
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.
FORMULA
EXAMPLE
0.1932888316392827389646154593552381142952702225292219922936048103344...
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Vaclav Kotesovec, Dec 03 2015
STATUS
approved