OFFSET
0,2
COMMENTS
Equal to the derivative eta'(1) of the Dirichlet eta function eta(s) = Sum_{k>=1} (-1)^(k-1)/k^s = (1 - 2^(1-s))*zeta(s) at s = 1. - Jonathan Sondow, Dec 28 2011
REFERENCES
A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series, Vol. 1, Overseas Publishers Association, Amsterdam, 1986, p. 746, section 5.5.1, formula 3.
LINKS
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.
Wikipedia, Dirichlet eta function.
FORMULA
Equals gamma*log(2) - log(2)^2/2.
Equals -Sum_{k>=1} psi(k)/(k*2^k), where psi(x) is the digamma function. - Amiram Eldar, Sep 12 2022
EXAMPLE
0.15986890374243097175694787032491657049622202375645874267082452963965...
MAPLE
gamma*log(2)-log(2)^2/2 ; evalf(%) ; # R. J. Mathar, Jun 10 2024
MATHEMATICA
RealDigits[EulerGamma*Log[2] - Log[2]^2/2, 10, 100][[1]] (* Amiram Eldar, Sep 12 2022 *)
RealDigits[Limit[Derivative[1][DirichletEta][x], x -> 1], 10, 110][[1]] (* Eric W. Weisstein, Jan 08 2024 *)
PROG
(PARI) Euler*log(2)-log(2)^2/2 \\ Charles R Greathouse IV, Mar 28 2012
CROSSREFS
KEYWORD
cons,nonn
AUTHOR
Benoit Cloitre, Mar 07 2004
STATUS
approved