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%I #16 Feb 10 2022 11:46:18
%S 3,4,2,9,8,1,6,2,6,0,0,2,3,0,5,6,0,6,5,0,2,2,4,1,1,5,8,5,6,5,5,8,6,0,
%T 5,4,0,4,5,2,3,7,6,2,0,0,1,2,0,7,1,0,3,8,9,8,4,8,2,0,0,5,2,0,9,7,4,0,
%U 4,4,4,2,8,3,5,9,4,8,1,6,1,2,1,1,8,7,4,1,9,7,2,3,8,7,3,5,3,4,5,1,6,7,7,4,2
%N Decimal expansion of Sum_{k>=2} 1/(k*(log k)^5).
%C Quintic 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_, Feb 08 2022
%H Wikipédia, <a href="https://fr.wikipedia.org/wiki/Série_de_Bertrand">Série de Bertrand</a> (in French).
%e 3.4298162600230560650224115856558605404523762001207...
%t (* Computation needs a few minutes *) digits = 105; NSum[ 1/(n*Log[n]^5), {n, 2, Infinity}, NSumTerms -> 1500000, WorkingPrecision -> digits + 10, Method -> {"EulerMaclaurin", Method -> {"NIntegrate", "MaxRecursion" -> 10}}] // RealDigits[#, 10, digits] & // First (* _Jean-François Alcover_, Feb 12 2013 *)
%Y Cf. A115563 (q=2), A145419 (q=3), A145420 (q=4).
%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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