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

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

Author

Bernard Schott, Oct 02 2021

Keywords

• nonn
• cons
• more

Comments

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.

Examples

			1.84825451761121890381193149396...
		

References

• 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.

Crossrefs

Cf. A001008, A002805, A115563.

Programs

Mathematica

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

Extensions

More terms from Amiram Eldar, Oct 02 2021