A145420 Decimal expansion of Sum_{k>=2} 1/(k*(log k)^4).

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, 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, 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

Offset

1,1

Comments

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.

Bertrand series Sum_{n>=2} 1/(n*log(n)^q) is convergent iff q > 1. - Bernard Schott, Jan 22 2022

Links

Table of n, a(n) for n=1..105.

R. J. Mathar, The series limit of sum_k 1/[k log k (log log k)^2], arXiv:0902.0789 [math.NA], 2009-2021, last sentence.

Wikipédia, Série de Bertrand (in French).

Example

2.5591197429867314185720209703107629336191781563668...

Mathematica

(* 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 *)

Crossrefs

Cf. A115563 (q=2), A145419 (q=3), A145421 (q=5).

Sequence in context: A168071 A361861 A330447 * A336257 A284169 A152781

Adjacent sequences: A145417 A145418 A145419 * A145421 A145422 A145423

Keyword

cons,nonn

Author

R. J. Mathar, Feb 08 2009

Extensions

More terms from Jean-François Alcover, Feb 12 2013

Status

approved