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%I #22 Jan 27 2022 20:49:23
%S 2,5,5,9,1,1,9,7,4,2,9,8,6,7,3,1,4,1,8,5,7,2,0,2,0,9,7,0,3,1,0,7,6,2,
%T 9,3,3,6,1,9,1,7,8,1,5,6,3,6,6,8,7,9,4,8,7,1,7,0,6,7,9,7,0,7,9,1,4,6,
%U 5,9,0,9,8,1,6,6,1,7,1,7,6,6,5,9,3,7,9,5,9,9,2,4,9,0,3,2,1,3,8,3,5,5,4,5,8
%N Decimal expansion of Sum_{k>=2} 1/(k*(log k)^4).
%C Quartic 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 Bertrand series Sum_{n>=2} 1/(n*log(n)^q) is convergent iff q > 1. - _Bernard Schott_, Jan 22 2022
%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, last sentence.
%H Wikipédia, <a href="https://fr.wikipedia.org/wiki/Série_de_Bertrand">Série de Bertrand</a> (in French).
%e 2.5591197429867314185720209703107629336191781563668...
%t (* Computation needs a few minutes *) digits = 105; NSum[ 1/(n*Log[n]^4), {n, 2, Infinity}, NSumTerms -> 800000, WorkingPrecision -> digits + 5, Method -> {"EulerMaclaurin", Method -> {"NIntegrate", "MaxRecursion" -> 10}}] // RealDigits[#, 10, digits] & // First (* _Jean-François Alcover_, Feb 12 2013 *)
%Y Cf. A115563 (q=2), A145419 (q=3), A145421 (q=5).
%K cons,nonn
%O 1,1
%A _R. J. Mathar_, Feb 08 2009
%E More terms from _Jean-François Alcover_, Feb 12 2013
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