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

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, 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, 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

Offset: 1

Author

Bernard Schott, May 13 2022

Keywords

• nonn
• cons

Comments

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.

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.

Note that if u(n) = 1 / n^2, n>0, then also v(n) = n^2 / (n^4 + 1).

Examples

			1.12852792472431008541205863...
		

References

• Jean-Marie Monier, Analyse, Tome 3, 2ème année, MP.PSI.PC.PT, Dunod, 1997, Exercice 3.2.3 pp. 249 and 444.

Crossrefs

Cf. A013661, A013662, A354051.

Programs

Maple

evalf(sum(n^2/(1+n^4),n=1..infinity),110);
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

Mathematica

RealDigits[Re[Sum[n^2/(n^4 + 1), {n, 1, Infinity}]], 10, 100][[1]] (* Amiram Eldar, May 13 2022 *)

PARI

sumpos(n=1, n^2/(n^4 + 1)) \\ Michel Marcus, May 16 2022

Formula

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

Equals 1/2 + Sum_{j>=0} (-1)^j*Zeta(2+4*j) = 1/2 + A013661 - A013664 + A013668 -.... - R. J. Mathar, May 20 2022