A263193 Decimal expansion of Sum_{n >= 1} sin(n)/sqrt(n).

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

G. C. Greubel, Table of n, a(n) for n = 1..10000

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

Cf. A096444, A113024, A263192.

Sequence in context: A222471 A180858 A365323 * A094885 A240199 A094728

Adjacent sequences: A263190 A263191 A263192 * A263194 A263195 A263196

Keyword

nonn,cons

Author

Iaroslav V. Blagouchine, Oct 11 2015

Status

approved