A145421 Decimal expansion of Sum_{k>=2} 1/(k*(log k)^5).

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, 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, 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

Offset

1,1

Comments

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.

Bertrand series Sum_{n>=2} 1/(n*log(n)^q) is convergent iff q > 1. - Bernard Schott, Feb 08 2022

Links

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

Wikipédia, Série de Bertrand (in French).

Example

3.4298162600230560650224115856558605404523762001207...

Mathematica

(* 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 *)

Crossrefs

Cf. A115563 (q=2), A145419 (q=3), A145420 (q=4).

Sequence in context: A215175 A091477 A075593 * A338245 A352417 A286681

Adjacent sequences: A145418 A145419 A145420 * A145422 A145423 A145424

Keyword

cons,nonn

Author

R. J. Mathar, Feb 08 2009

Extensions

More terms from Jean-François Alcover, Feb 12 2013

Status

approved