A168218 Decimal expansion of the Sum_{k=2..infinity} 1/(k^2*log(k)).

6, 0, 5, 5, 2, 1, 7, 8, 8, 8, 8, 2, 6, 0, 0, 4, 4, 7, 6, 9, 9, 5, 4, 9, 0, 0, 5, 2, 0, 7, 2, 4, 0, 4, 4, 7, 3, 0, 3, 2, 3, 8, 8, 9, 8, 4, 5, 5, 0, 6, 5, 7, 8, 3, 3, 1, 1, 1, 4, 7, 5, 9, 0, 4, 2, 0, 6, 8, 9, 4, 1, 1, 9, 7, 8, 0, 8, 8, 6, 8, 1, 7, 6, 1, 1, 8, 3, 1, 2, 8, 4, 1, 9, 3, 0, 8, 9, 4, 0, 1, 9, 8, 6, 9, 9

Offset: 0

Author

R. J. Mathar, Nov 20 2009

Keywords

• cons
• nonn

Comments

Also the value of the integral of the fractional part of the Riemann zeta function from 2 to infinity. - Alexander Bock, Apr 01 2014

Examples

			equals 1/(4*log(2))+1/(9*log(3))+1/(16*log(4))+ .... + = 0.605521788882600447699549005207240447303238898..
		

Links

• R. P. Boas, Jr, Partial sums of Infinite Series, and How They Grow, Am. Math. Monthly 84 (4) (1977) 237-235 [MR0440240].

Crossrefs

Cf. A099769, A115563, A257812.

Programs

Mathematica

(* Computation needs a few minutes for 105 digits *) digits = 105; NSum[ 1/(n^2*Log[n]), {n, 2, Infinity}, NSumTerms -> 500000, WorkingPrecision -> digits + 5, Method -> {"EulerMaclaurin", Method -> {"NIntegrate", "MaxRecursion" -> 12}}] // RealDigits[#, 10, digits] & // First (* Jean-François Alcover, Feb 11 2013 *)
RealDigits[NIntegrate[Zeta[x] - 1, {x, 2, Infinity}, WorkingPrecision->110], 10, 100] (* Alexander Bock, Apr 01 2014 *)

PARI

intnum(x=2,[oo,log(2)],zeta(x)-1) \\ Charles R Greathouse IV, Apr 01 2014

PARI

suminf(k=2,1/k^2/log(k)) \\ Charles R Greathouse IV, Apr 01 2014

Formula

Equals Integral_{x>=2} (zeta(x) - 1) dx.

Extensions

More terms from Jean-François Alcover, Feb 11 2013