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

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, 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, 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

Offset: 1

Author

R. J. Mathar, Feb 08 2009

Keywords

• cons
• nonn

Comments

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.

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

Examples

			2.0658865388841352509031422416437738180869752069383...
		

Links

• R. J. Mathar, The series limit of sum_k 1/[k log k (log log k)^2], arXiv:0902.0789 [math.NA], 2009-2021, constant D^(3).
• Wikipédia, Série de Bertrand (in French).

Crossrefs

Cf. A115563, A145420, A145421, A354917.

Programs

Mathematica

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 *)
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 *)

Extensions

More terms from Jean-François Alcover, Feb 11 2013

More digits from Vaclav Kotesovec, Jun 11 2022