cp's OEIS Frontend

A145419 Decimal expansion of Sum_{k>=2} 1/(k*(log k)^3).

%I A145419 #32 Jun 12 2022 06:36:42
%S A145419 2,0,6,5,8,8,6,5,3,8,8,8,4,1,3,5,2,5,0,9,0,3,1,4,2,2,4,1,6,4,3,7,7,3,
%T A145419 8,1,8,0,8,6,9,7,5,2,0,6,9,3,8,3,4,7,0,7,3,4,6,3,2,4,3,6,0,2,4,1,6,8,
%U A145419 0,7,4,0,1,3,7,7,6,5,1,5,8,6,5,5,2,6,7,3,8,2,7,3,1,4,3,0,1,3,8,8,7,7,1,8,8
%N A145419 Decimal expansion of Sum_{k>=2} 1/(k*(log k)^3).
%C A145419 Cubic analog of A115563. Evaluated by direct summation of the first 160 terms and accumulating the remainder with the 5 nontrivial terms in the Euler-Maclaurin expansion.
%C A145419 Theorem: Bertrand series Sum_{n>=2} 1/(n*log(n)^q) is convergent iff q > 1 (for q = 2, 4, 5 see respectively A115563, A145420, A145421). - _Bernard Schott_, Oct 23 2021
%H A145419 R. J. Mathar, <a href="http://arxiv.org/abs/0902.0789">The series limit of sum_k 1/[k log k (log log k)^2]</a>, arXiv:0902.0789 [math.NA], 2009-2021, constant D^(3).
%H A145419 Wikipédia, <a href="https://fr.wikipedia.org/wiki/Série_de_Bertrand">Série de Bertrand</a> (in French).
%e A145419 2.0658865388841352509031422416437738180869752069383...
%t A145419 digits = 50; NSum[ 1/(n*Log[n]^3), {n, 2, Infinity}, NSumTerms -> 10000, WorkingPrecision -> digits + 10] // RealDigits[#, 10, digits] & // First (* _Jean-François Alcover_, Feb 11 2013 *)
%t A145419 alfa = 3; maxiter = 20; nn = 10000; bas = Sum[1/(k*Log[k]^alfa), {k, 2, nn}] + 1/((alfa - 1)*Log[nn + 1/2]^(alfa - 1)); sub = 0; Do[sub = sub + 1/4^s/(2*s + 1)! * NSum[(D[1/(x*Log[x]^alfa), {x, 2 s}]) /. x -> k, {k, nn + 1, Infinity}, WorkingPrecision -> 120, NSumTerms -> 100000, PrecisionGoal -> 120, Method -> {"NIntegrate", "MaxRecursion" -> 100}]; Print[bas - sub], {s, 1, maxiter}] (* _Vaclav Kotesovec_, Jun 11 2022 *)
%Y A145419 Cf. A115563, A145420, A145421, A354917.
%K A145419 cons,nonn
%O A145419 1,1
%A A145419 _R. J. Mathar_, Feb 08 2009
%E A145419 More terms from _Jean-François Alcover_, Feb 11 2013
%E A145419 More digits from _Vaclav Kotesovec_, Jun 11 2022