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

Author

Benoit Cloitre, Mar 07 2004

Keywords

• cons
• nonn

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

Examples

			0.15986890374243097175694787032491657049622202375645874267082452963965...
		

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

• 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.

Crossrefs

Cf. A001620, A099769, A265162, A354295.

Programs

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 *)

PARI

Euler*log(2)-log(2)^2/2 \\ Charles R Greathouse IV, Mar 28 2012

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