%I #17 Apr 27 2023 01:49:00

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

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

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

%C 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"

%D 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.

%H R. J. Mathar, <a href="http://arxiv.org/abs/0902.0789">The series limit of sum_k 1/[k log k (log log k)^2]</a>, arXiv:0902.0789 [math.NA].

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ConvergentSeries.html">Convergent Series</a>

%e 38.4067...

%t (* 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 *)

%Y Cf. A115563.

%K cons,more,nonn

%O 2,1

%A _Robert G. Wilson v_, May 16 2006

%E Corrected the least significant digit and added 11 more digits. _R. J. Mathar_, Feb 03 2009

%E Name spelling and 3 least significant digits corrected by _R. J. Mathar_, Jul 07 2009