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

%I A348763 #24 May 04 2025 14:43:56
%S A348763 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,
%T A348763 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,
%U A348763 0,1,1,3,2,3,1,9,0,5,7,2,1,3,1,0,9
%N A348763 Decimal expansion of Sum_{n>=1} ((-1)^(n+1)*n)/(n+1)^2.
%H A348763 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Dilogarithm.html">Dilogarithm</a>
%H A348763 <a href="/index/Z#zeta_function">Index entries for zeta function</a>.
%F A348763 Equals Pi^2/12-log(2).
%F A348763 Equals Sum_{k>=2} (zeta(k)-zeta(k+1))/2^k. - _Amiram Eldar_, Mar 20 2022
%F A348763 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
%e A348763 0.12931985286416790881897546186483602653397481624314396474709910519161011...
%t A348763 RealDigits[Pi^2/12 - Log[2], 10, 100][[1]] (* _Amiram Eldar_, Nov 30 2021 *)
%o A348763 (SageMath) (pi^2/12-log(2)).n(digits=100)
%o A348763 (PARI) -sumalt(n=1, (-1)^n*n/(n+1)^2) \\ _Charles R Greathouse IV_, Nov 01 2021
%o A348763 (PARI) Pi^2/12-log(2) \\ _Charles R Greathouse IV_, Nov 01 2021
%o A348763 (Python)
%o A348763 from scipy.special import zeta
%o A348763 from math import log
%o A348763 int(''.join(n for n in list(str(zeta(2)/2-log(2)))[2:-2]))
%o A348763 (Python)
%o A348763 int(str(sum((-1)**(n+1)*n/(n+1)**2 for n in range(1,5000000)))[2:-2])
%Y A348763 Cf. A006752, A072691, A111003, A153071, A175571, A181983, A300707.
%K A348763 nonn,cons
%O A348763 0,2
%A A348763 _Dumitru Damian_, Oct 31 2021