OFFSET
2,1
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"
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].
Eric Weisstein's World of Mathematics, Convergent Series
EXAMPLE
38.4067...
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 *)
CROSSREFS
KEYWORD
AUTHOR
Robert G. Wilson v, May 16 2006
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
STATUS
approved