A354051 -id:A354051 - cp's OEIS frontend

A354052 Decimal expansion of Sum_{k>=0} 1 / (k^6 + 1).

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

Offset: 1

Vaclav Kotesovec, May 16 2022

			1.517100734033216426152907644902413858062211322529844672847634899037901...

Cf. A002604, A113319, A354051, A354053.

evalf(1/2 + (coth(Pi) + (sinh(Pi) + sqrt(3)*sin(sqrt(3)*Pi)) / (cosh(Pi) - cos(sqrt(3)*Pi)))*Pi/6, 100);

RealDigits[Chop[N[Sum[1/(k^6 + 1), {k, 0, Infinity}], 105]]][[1]]

PARI
```
sumpos(k=0, 1/(k^6 + 1))
```

Equals 1/2 + (coth(Pi) + (sinh(Pi) + sqrt(3)*sin(sqrt(3)*Pi)) / (cosh(Pi) - cos(sqrt(3)*Pi)))*Pi/6.

Equal 3/2 + Sum_{k>=1} (-1)^(k+1) * (zeta(6*k)-1). - Amiram Eldar, May 20 2022

A354053 Decimal expansion of Sum_{k>=0} 1 / (k^8 + 1).

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, May 16 2022

			1.504062133314799511292905417451127075245414363820351975458635357818812...

Cf. A060890, A113319, A354051, A354052.

evalf(1/2 + ((sqrt(2 + sqrt(2))*sinh(sqrt(2 + sqrt(2))*Pi) + sqrt(2 - sqrt(2))*sin(sqrt(2 - sqrt(2))*Pi)) / (cosh(sqrt(2 + sqrt(2))*Pi) - cos(sqrt(2 - sqrt(2))*Pi)) + (sqrt(2 + sqrt(2))*sin(sqrt(2 + sqrt(2))*Pi) + sqrt(2 - sqrt(2))*sinh(sqrt(2 - sqrt(2))*Pi)) / (cosh(sqrt(2 - sqrt(2))*Pi) - cos(sqrt(2 + sqrt(2))*Pi))) * Pi/8, 100);

RealDigits[Chop[N[Sum[1/(k^8 + 1), {k, 0, Infinity}], 105]]][[1]]

PARI
```
sumpos(k=0, 1/(k^8 + 1))
```

Equals 1/2 + ((sqrt(2 + sqrt(2))*sinh(sqrt(2 + sqrt(2))*Pi) + sqrt(2 - sqrt(2))*sin(sqrt(2 - sqrt(2))*Pi)) / (cosh(sqrt(2 + sqrt(2))*Pi) - cos(sqrt(2 - sqrt(2))*Pi)) + (sqrt(2 + sqrt(2))*sin(sqrt(2 + sqrt(2))*Pi) + sqrt(2 - sqrt(2))*sinh(sqrt(2 - sqrt(2))*Pi)) / (cosh(sqrt(2 - sqrt(2))*Pi) - cos(sqrt(2 + sqrt(2))*Pi))) * Pi/8.

Equal 3/2 + Sum_{k>=1} (-1)^(k+1) * (zeta(8*k)-1). - Amiram Eldar, May 20 2022

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

Original entry on oeis.org

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

Bernard Schott, May 13 2022

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).

			1.12852792472431008541205863...

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

Cf. A013661, A013662, A354051.

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

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

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

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

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A354052 Decimal expansion of Sum_{k>=0} 1 / (k^6 + 1).

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A354053 Decimal expansion of Sum_{k>=0} 1 / (k^8 + 1).

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

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