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A263193
Decimal expansion of Sum_{n >= 1} sin(n)/sqrt(n).
6
1, 0, 4, 3, 9, 8, 2, 1, 0, 2, 8, 4, 9, 1, 6, 1, 5, 2, 7, 4, 5, 3, 2, 9, 4, 8, 7, 2, 5, 8, 6, 7, 5, 0, 4, 5, 9, 7, 9, 0, 9, 0, 7, 1, 4, 4, 7, 2, 2, 6, 1, 2, 2, 0, 3, 7, 4, 8, 9, 5, 2, 8, 5, 8, 7, 7, 0, 6, 6, 9, 0, 8, 5, 8, 6, 0, 0, 3, 2, 4, 2, 1, 5, 7, 2, 9, 0, 1, 0, 9, 2, 4, 7, 7, 2, 2, 0, 1, 2, 7, 5, 5, 7, 3, 7, 1, 9, 3, 7
OFFSET
1,3
COMMENTS
A slowly convergent series. It may be efficiently computed via the Hurwitz zeta-function (see formula below).
LINKS
Iaroslav V. Blagouchine, A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments and some related summations, Journal of Number Theory (Elsevier), vol. 148, pp. 537-592 & vol. 151, pp. 276-277, 2015. arXiv version, arXiv:1401.3724 [math.NT].
FORMULA
(Zeta(1/2, 1/(2*Pi)) - Zeta(1/2, 1-1/(2*Pi)))/2, see formula (26) in the reference.
EXAMPLE
1.043982102849161527453294872586750459790907144722612...
MAPLE
evalf(1/2*(Zeta(0, 1/2, 1/(2*Pi)) - Zeta(0, 1/2, 1-1/(2*Pi))), 120);
MATHEMATICA
N[(Zeta[1/2, 1/(2*Pi)] - Zeta[1/2, 1 - 1/(2*Pi)])/2, 200]
RealDigits[Re[(1/2)*I*(PolyLog[1/2, E^(-I)] - PolyLog[1/2, E^I])], 10, 109][[1]] (* Vaclav Kotesovec, Oct 31 2015 *)
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
STATUS
approved