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Decimal expansion of Sum_{n>=1} 1/(n*H(n)^2) where H(n) is the n-th harmonic number.
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%I #39 Oct 21 2021 15:14:07

%S 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

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

%C 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.

%C 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.

%C This sequence gives this limit when q = 2.

%D Xavier Gourdon, Analyse, Les Maths en tête, Exercice 5, page 213, Ellipses, 1994.

%D 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.

%e 1.84825451761121890381193149396...

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

%Y Cf. A001008, A002805, A115563.

%K nonn,cons,more

%O 1,2

%A _Bernard Schott_, Oct 02 2021

%E More terms from _Amiram Eldar_, Oct 02 2021