A348763 Decimal expansion of Sum_{n>=1} ((-1)^(n+1)*n)/(n+1)^2.

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

Offset

0,2

Links

Table of n, a(n) for n=0..84.

Eric Weisstein's World of Mathematics, Dilogarithm.

Index entries for zeta function.

Formula

Equals Pi^2/12-log(2).

Equals Sum_{k>=2} (zeta(k)-zeta(k+1))/2^k. - Amiram Eldar, Mar 20 2022

Equals Integral_{x >= 0} x/(1 + exp(x))^2 dx = (1/2) * Integral_{x >= 0} x*(x - 2)*exp(x)/(1 + exp(x))^2 dx . - Peter Bala, Apr 26 2025

Equals -1/4 + Integral_{x=0..1} x * PolyLog(2, 1/1(x+1)) dx. - Amiram Eldar, Nov 06 2025

Equals Sum_{i,j >= 1} (-1)^(i+j) / (i+j)^2. - Amiram Eldar, Jul 04 2026

Example

0.12931985286416790881897546186483602653397481624314396474709910519161011...

Mathematica

RealDigits[Pi^2/12 - Log[2], 10, 100][[1]] (* Amiram Eldar, Nov 30 2021 *)

Prog

(SageMath) (pi^2/12-log(2)).n(digits=100)
(PARI) -sumalt(n=1, (-1)^n*n/(n+1)^2) \\ Charles R Greathouse IV, Nov 01 2021
(PARI) Pi^2/12-log(2) \\ Charles R Greathouse IV, Nov 01 2021
(Python)
from scipy.special import zeta
from math import log
int(''.join(n for n in list(str(zeta(2)/2-log(2)))[2:-2]))
(Python)
int(str(sum((-1)**(n+1)*n/(n+1)**2 for n in range(1, 5000000)))[2:-2])

Crossrefs

Cf. A006752, A072691, A111003, A153071, A175571, A181983, A300707.

Sequence in context: A179451 A124918 A395472 * A011240 A021345 A375771

Adjacent sequences: A348760 A348761 A348762 * A348764 A348765 A348766

Keyword

nonn,cons,changed

Author

Dumitru Damian, Oct 31 2021

Status

approved