A115563 -id:A115563 - cp's OEIS frontend

A363632 Decimal expansion of Sum_{k>=2} 1/(k* log(k)^(3/2)).

2, 9, 3, 7, 6, 6, 3, 6, 3, 7, 9, 0, 1, 2, 3, 1, 7, 7

Offset: 1

R. J. Mathar, Jun 12 2023

			2.93766363790123177...

Cf. A115563 (expo 2), A363633 (expo 5/2), A145419 (expo 3), A145420 (expo 4).

A363633 Decimal expansion of Sum_{k>=2} 1/(k* log(k)^(5/2)).

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Jun 12 2023

			1.98346354715623939118...

Cf. A363632 (expo 3/2), A115563 (expo 2), A145419 (expo 3), A145420 (expo 4).

A347145 Decimal expansion of Sum_{n>=1} 1/(n*H(n)^2) where H(n) is the n-th harmonic number.

Original entry on oeis.org

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

Offset: 1

Bernard Schott, Oct 02 2021

Theorem: If u(n) is a series with positive terms such that u(n) -> 0 when n -> oo and that is divergent, i.e., Sum_{n>=0} u(n) = oo, let S(n) = Sum_{k=0..n} u(k) then, the series of term v(n) = u(n)/S(n)^q is convergent iff q>1.
The simplest application is for u(n) = 1/n, S(n) = H(n) = 1 + 1/2 + ... + 1/n, then the series of term w(n) = 1/(n*H(n)^q) is convergent iff q>1.
This sequence gives this limit when q = 2.

			1.84825451761121890381193149396...

Xavier Gourdon, Analyse, Les Maths en tête, Exercice 5, page 213, Ellipses, 1994.
J. Lelong-Ferrand and J. M. Arnaudiès, Cours de Mathématiques, Tome 2, Analyse, 4ème édition, Classes préparatoires, 1er cycle universitaire, Exercice 21, p. 599, Dunod Université, 1977.

Cf. A001008, A002805, A115563.

RealDigits[N[Sum[1/(n*HarmonicNumber[n]^2), {n, 1, Infinity}], 33], 10, 30][[1]] (* Amiram Eldar, Oct 02 2021 *)

More terms from Amiram Eldar, Oct 02 2021

A352473 Decimal expansion of Sum_{k>=2} (log(k!) / k!).

Original entry on oeis.org

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

Offset: 0

Bernard Schott, Mar 17 2022

Sum_{k>=2} (log(k) / k) is divergent but here, this series is convergent. If v(k) = log(k!) / k!, we have 0 <= v(k) <= w(k) = k^2/k! with w(k) that is convergent, hence, this positive series is convergent.

			0.8286471276718785080389169468558478918...

J.-M. Monier, Analyse, Tome 3, 2ème année, MP.PSI.PC.PT, Dunod, 1997, Exercice 3.2.13.b p. 250.

Cf. A099769, A115563, A168218, A257812.

Maple
```
sum(log(n!)/n!, n=2..infinity);
```

sumpos(k=2, log(k!)/k!) \\ Michel Marcus, Mar 17 2022

Equals Sum_{k>=2} (log(k!) / k!).

cp's OEIS Frontend

A363632 Decimal expansion of Sum_{k>=2} 1/(k* log(k)^(3/2)).

Views

Author

Keywords

Examples

Crossrefs

A363633 Decimal expansion of Sum_{k>=2} 1/(k* log(k)^(5/2)).

Views

Author

Keywords

Examples

Crossrefs

A347145 Decimal expansion of Sum_{n>=1} 1/(n*H(n)^2) where H(n) is the n-th harmonic number.

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

Mathematica

Extensions

A352473 Decimal expansion of Sum_{k>=2} (log(k!) / k!).

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

Maple

PARI

Formula