A118582 Decimal expansion of Sum_{k>=3} 1/(k log k (log log k)^2).

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

Offset: 2

Author

Robert G. Wilson v, May 16 2006

Keywords

• cons
• more
• nonn
• hard

Comments

"The series [see title] converges to 38.43... so slowly that it requires 10^(3.14*10^86) terms to give two-decimal accuracy"

example 16 - "The series Sum_{k=3..inf} 1/(k log k (log log k)) diverges, but the partial sums exceed 10 only after a googolplex of terms have appeared"

Examples

			38.4067...
		

References

• Daniel Zwillinger, Editor, CRC Standard Mathematical Tables and Formulae, 31st Edition, Chapman & Hall/CRC, Boca Raton, 1.3.9 Miscellaneous Sums and Series, example 15, page 42, 2003.

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.
• Eric Weisstein's World of Mathematics, Convergent Series

Crossrefs

Cf. A115563.

Programs

Mathematica

(* Computation needs a few minutes *) digits = 15; m0 = 10^6; dm = 10^5; Clear[f]; f[m_] := f[m] = Sum[ 1/(k*Log[k]*Log[Log[k]]^2) // N[#, digits+2]&, {k, 3, m}] + 1/Log[Log[m + 1/2]] // RealDigits[#, 10, digits+2]& // First; f[m0]; f[m = m0 + dm]; While[f[m] != f[m - dm], m = m + dm]; f[m][[1 ;; digits]] (* Jean-François Alcover, Mar 07 2013 *)

Extensions

Corrected the least significant digit and added 11 more digits. R. J. Mathar, Feb 03 2009

Name spelling and 3 least significant digits corrected by R. J. Mathar, Jul 07 2009