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

1,2

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.

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.

Links

Table of n, a(n) for n=1..30.

Example

1.84825451761121890381193149396...

Mathematica

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

Crossrefs

Cf. A001008, A002805, A115563.

Sequence in context: A155889 A275712 A199266 * A205383 A083948 A021848

Adjacent sequences: A347142 A347143 A347144 * A347146 A347147 A347148

Keyword

nonn,cons,more

Author

Bernard Schott, Oct 02 2021

Extensions

More terms from Amiram Eldar, Oct 02 2021

Status

approved