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Decimal expansion of Sum_{k>=2} 1/(k*(log k)^3).
6

%I #32 Jun 12 2022 06:36:42

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

%C 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 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 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 Wikipédia, <a href="https://fr.wikipedia.org/wiki/Série_de_Bertrand">Série de Bertrand</a> (in French).

%e 2.0658865388841352509031422416437738180869752069383...

%t 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 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 Cf. A115563, A145420, A145421, A354917.

%K cons,nonn

%O 1,1

%A _R. J. Mathar_, Feb 08 2009

%E More terms from _Jean-François Alcover_, Feb 11 2013

%E More digits from _Vaclav Kotesovec_, Jun 11 2022