A263192 Decimal expansion of Sum_{n >= 1} cos(n)/sqrt(n), negated.

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

Offset

0,2

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 = 0..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

-0.1941089350921820497391492449281947266353205526340478...

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)*(PolyLog[1/2, E^(-I)] + PolyLog[1/2, E^I])], 10, 109][[1]] (* Vaclav Kotesovec, Oct 31 2015 *)

Prog

(PARI) zetahurwitz(1/2, 1/Pi/2)/2 + zetahurwitz(1/2, 1-1/Pi/2)/2 \\ Charles R Greathouse IV, Jan 30 2018

Crossrefs

Cf. A113024, A121225, A263193.

Sequence in context: A021918 A112146 A056897 * A355282 A270309 A010158

Adjacent sequences: A263189 A263190 A263191 * A263193 A263194 A263195

Keyword

nonn,cons

Author

Iaroslav V. Blagouchine, Oct 11 2015

Status

approved