A084254 Decimal expansion of Sum_{k>=1} 1/(k*(exp(2*Pi*k)-1)).

0, 0, 1, 8, 7, 2, 6, 8, 2, 4, 4, 9, 7, 6, 8, 5, 4, 6, 1, 1, 5, 6, 3, 8, 5, 7, 9, 4, 7, 9, 9, 6, 1, 3, 9, 8, 8, 6, 9, 1, 6, 2, 8, 9, 5, 6, 5, 2, 6, 1, 9, 5, 6, 3, 8, 4, 1, 3, 3, 1, 5, 7, 4, 5, 3, 7, 8, 8, 4, 3, 1, 9, 5, 1, 7, 0, 9, 8, 0, 2, 2, 6, 7, 5, 1, 7, 0, 7, 2, 7, 8, 4, 0, 2, 4, 5, 6, 7, 9, 7, 9, 9, 8, 7

Offset: 0

Author

Benoit Cloitre, Jun 21 2003

Keywords

• cons
• nonn
• easy

Examples

			0.00187268244976854611563857947996139886916289565261...
		

References

• Bruce C. Berndt, Ramanujan Notebook part II, Infinite series, Springer Verlag, 1989, pp. 280-281.

Links

• Steven R. Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020-2022, p. 6.
• Simon Plouffe, Identities inspired by Ramanujan Notebooks (part 2), April 2006.
• Linas Vepštas, On Plouffe's Ramanujan identities, The Ramanujan Journal, Vol. 27 (2012), pp. 387-408; arXiv preprint, arXiv:math/0609775 [math.NT], 2006-2010.

Crossrefs

Cf. A000203, A003881, A068465, A094640.

Cf. A255695 (S(1,1)), A255697 (S(1,4)), A255698 (S(3,1)), A255699 (S(3,2)), A255700 (S(3,4)), A255701 (S(5,1)), A255702 (S(5,2)), A255703 (S(5,4)).

Programs

Mathematica

digits = 104; S[1, 2] = NSum[1/(n*(Exp[2*Pi*n] - 1)), {n, 1, Infinity}, WorkingPrecision -> digits+10, NSumTerms -> digits]; RealDigits[S[1, 2], 10, digits] // First (* Jean-François Alcover, Mar 02 2015 *)
Join[{0,0},RealDigits[Log[4/Pi]/4 - Pi/12 + Log[Gamma[3/4]], 10, 100][[1]]] (* Amiram Eldar, May 21 2022 *)

PARI

1/4*log(4/Pi)-Pi/12+log(gamma(3/4))

Formula

Equals log(4/Pi)/4 - Pi/12 + log(Gamma(3/4)).

From Jean-François Alcover, Mar 02 2015: (Start)

This is the case k=1, m=2 of the Plouffe sum S(k,m) = Sum_{n >= 1} 1/(n^k*(exp(m*Pi*n)-1)).

Pi = 72*S(1,1) - 96*S(1,2) + 24*S(1,4). (End)

Equals Sum_{k>=1} sigma(k)/(k*exp(2*Pi*k)). - Amiram Eldar, Jun 05 2023