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%I #56 Jan 11 2024 05:23:19
%S 1,5,9,8,6,8,9,0,3,7,4,2,4,3,0,9,7,1,7,5,6,9,4,7,8,7,0,3,2,4,9,1,6,5,
%T 7,0,4,9,6,2,2,2,0,2,3,7,5,6,4,5,8,7,4,2,6,7,0,8,2,4,5,2,9,6,3,9,6,5,
%U 7,0,0,2,1,8,4,0,2,9,0,0,4,6,5,9,5,5,5,0,3,4,0,3,2,0,4,6,1,8,8,2,9,4,6,3
%N Decimal expansion of Sum_{k>=1} ((-1)^k*log(k)/k).
%C 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
%D 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.
%H Henri Cohen, Fernando Rodriguez Villegas, and Don Zagier, <a href="https://dx.doi.org/10.1080/10586458.2000.10504632">Convergence Acceleration of Alternating Series</a>, Exp. Math. 9 (1) (2000) 3-12.
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DirichletEtaFunction.html">Dirichlet Eta Function</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Dirichlet_eta_function">Dirichlet eta function</a>.
%F Equals gamma*log(2) - log(2)^2/2.
%F Equals -Sum_{k>=1} psi(k)/(k*2^k), where psi(x) is the digamma function. - _Amiram Eldar_, Sep 12 2022
%e 0.15986890374243097175694787032491657049622202375645874267082452963965...
%t RealDigits[EulerGamma*Log[2] - Log[2]^2/2, 10, 100][[1]] (* _Amiram Eldar_, Sep 12 2022 *)
%t RealDigits[Limit[Derivative[1][DirichletEta][x], x -> 1], 10, 110][[1]] (* _Eric W. Weisstein_, Jan 08 2024 *)
%o (PARI) Euler*log(2)-log(2)^2/2 \\ _Charles R Greathouse IV_, Mar 28 2012
%Y Cf. A001620, A099769, A265162, A354295.
%K cons,nonn
%O 0,2
%A _Benoit Cloitre_, Mar 07 2004
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