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Decimal expansion of the Plouffe sum S(5,2) = Sum_{n >= 1} 1/(n^5*(exp(2*Pi*n)-1)).
8

%I #21 Jun 05 2023 02:01:54

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

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

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

%N Decimal expansion of the Plouffe sum S(5,2) = Sum_{n >= 1} 1/(n^5*(exp(2*Pi*n)-1)).

%H Steven R. Finch, <a href="https://arxiv.org/abs/2001.00578">Errata and Addenda to Mathematical Constants</a>, arXiv:2001.00578 [math.HO], 2020-2022, p. 6.

%H Simon Plouffe, <a href="http://www.plouffe.fr/simon/inspired2.pdf">Identities inspired by Ramanujan Notebooks (part 2)</a>, April 2006.

%H Linas Vepštas, <a href="https://doi.org/10.1007/s11139-011-9335-9">On Plouffe's Ramanujan identities</a>, The Ramanujan Journal, Vol. 27 (2012), pp. 387-408; <a href="https://cyberleninka.org/article/n/534457.pdf">alternative link</a>; <a href="https://arxiv.org/abs/math/0609775">arXiv preprint</a>, arXiv:math/0609775 [math.NT], 2006-2010.

%F This is the case k=5, m=2 of S(k,m) = Sum_{n >= 1} 1/(n^k*(exp(m*Pi*n)-1)).

%F zeta(5) = 24*S(5,1) - (259/10)*S(5,2) - (1/10)*S(5,4).

%F Equals Sum_{k>=1} sigma_5(k)/(k^5*exp(2*Pi*k)). - _Amiram Eldar_, Jun 05 2023

%e 0.0018710456053012595148739514758105634303189628230875828656...

%t digits = 104; S[5, 2] = NSum[1/(n^5*(Exp[2*Pi*n] - 1)), {n, 1, Infinity}, WorkingPrecision -> digits+10, NSumTerms -> digits]; RealDigits[S[5, 2], 10, digits] // First

%Y Cf. A255695 (S(1,1)), A084254 (S(1,2)), A255697 (S(1,4)), A255698 (S(3,1)), A255699 (S(3,2)), A255700 (S(3,4)), A255701 (S(5,1)), A255703 (S(5,4)).

%Y Cf. A001160 (sigma_5), A013663 (zeta(5)).

%K nonn,cons,easy

%O -2,2

%A _Jean-François Alcover_, Mar 02 2015