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A354004 Decimal expansion of Sum_{n>0} n^2 / (n^4 + 1).

%I A354004 #31 May 20 2022 05:49:28
%S A354004 1,1,2,8,5,2,7,9,2,4,7,2,4,3,1,0,0,8,5,4,1,2,0,5,8,6,3,3,7,4,9,7,2,8,
%T A354004 4,3,3,6,8,6,4,2,6,7,9,8,3,9,2,6,8,1,8,3,4,9,5,6,6,3,3,9,4,2,2,5,6,1,
%U A354004 2,5,5,8,8,5,9,0,5,4,1,3,4,2,5,8,5,0,5,4,1,5,0,3,2,6,0,4
%N A354004 Decimal expansion of Sum_{n>0} n^2 / (n^4 + 1).
%C A354004 When u(n) is a sequence of positive terms and Sum_{n>0} u(n) converges, if v(n) = u(n) / (1 + u(n)^2), then Sum_{n>0} v(n) also converges.
%C A354004 The converse is false; for example, when u(n) = n^2 then Sum_{n>0} u(n) = oo, but Sum_{n>0} n^2 / (n^4 + 1) is convergent and the limit of this series v(n) is this constant.
%C A354004 Note that if u(n) = 1 / n^2, n>0, then also v(n) = n^2 / (n^4 + 1).
%D A354004 Jean-Marie Monier, Analyse, Tome 3, 2ème année, MP.PSI.PC.PT, Dunod, 1997, Exercice 3.2.3 pp. 249 and 444.
%F A354004 Equals Pi*(sin(sqrt(2)*Pi) - sinh(sqrt(2)*Pi)) / (2*sqrt(2)*(cos(sqrt(2)*Pi) - cosh(sqrt(2)*Pi))). - _Vaclav Kotesovec_, May 16 2022
%F A354004 Equals 1/2 + Sum_{j>=0} (-1)^j*Zeta(2+4*j) = 1/2 + A013661 - A013664 + A013668 -.... - _R. J. Mathar_, May 20 2022
%e A354004 1.12852792472431008541205863...
%p A354004 evalf(sum(n^2/(1+n^4),n=1..infinity),110);
%p A354004 evalf(Pi*(sin(sqrt(2)*Pi) - sinh(sqrt(2)*Pi)) / (2*sqrt(2)*(cos(sqrt(2)*Pi) - cosh(sqrt(2)*Pi))), 121); # _Vaclav Kotesovec_, May 16 2022
%t A354004 RealDigits[Re[Sum[n^2/(n^4 + 1), {n, 1, Infinity}]], 10, 100][[1]] (* _Amiram Eldar_, May 13 2022 *)
%o A354004 (PARI) sumpos(n=1, n^2/(n^4 + 1)) \\ _Michel Marcus_, May 16 2022
%Y A354004 Cf. A013661, A013662, A354051.
%K A354004 nonn,cons
%O A354004 1,3
%A A354004 _Bernard Schott_, May 13 2022