A091812 Decimal expansion of Sum_{k>=1} (-1)^k*log(k)/k.

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, 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, 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

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

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.21, p. 168.

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

Table of n, a(n) for n=0..103.

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

Cf. A001620, A099769, A265162, A354295.

Sequence in context: A247747 A308577 A217249 * A112678 A021171 A011493

Adjacent sequences: A091809 A091810 A091811 * A091813 A091814 A091815

Keyword

cons,nonn,changed

Author

Benoit Cloitre, Mar 07 2004

Status

approved