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

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

Table of n, a(n) for n=2..106.

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.

Example

38.40676809282178631849374770114467...

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 *)

Prog

(PARI)
/* procedure by Bill Allombert */
1/log(log(3))+sumnum(n=3, my(u=log(n), v=log(u), w=log1p(1/n), t=log1p(w/u)); (-t/(v+t)+1/(n*u*v))/v)

Crossrefs

Cf. A115563, A394929.

Sequence in context: A281516 A281630 A278957 * A356580 A086179 A185453

Adjacent sequences: A118579 A118580 A118581 * A118583 A118584 A118585

Keyword

cons,nonn

Author

Robert G. Wilson v, May 16 2006

Extensions

Least significant digit corrected and 11 more digits from R. J. Mathar, Feb 03 2009

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

a(17)-a(106) from Artur Jasinski, Apr 20 2026

Status

approved