

A115563


Decimal expansion of Sum_{n>=2} 1/(n*log(n)^2).


11



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

1,1


COMMENTS

Sum_{n>1} 1/(n*log(n)^2) is a tiny bit greater than (zeta(2))^(3/2) = (Pi^2 / 6)^(3/2) = 2.109709908063657....  Daniel Forgues, Mar 30 2012
From Bernard Schott, Oct 03 2021: (Start)
Theorem: Bertrand series Sum_{n>=2} 1/(n*log(n)^q) is convergent iff q > 1 (for q = 3, 4, 5 see respectively A145419, A145420, A145421).
As H(n) ~ log(n), compare with A347145. (End)


LINKS

Table of n, a(n) for n=1..141.
John V. Baxley, Euler's constant, Taylor's formula, and slowly converging series, Math. Mag. 65 (1992), 302313.
Bart Braden, Calculating sums of infinite series, Am. Math. Monthly 99 (1992) 649655.
David Broadhurst, Re: need help about 2 constants, primeforum, Mar 10 2006.
Pierre CAMI, David Broadhurst, Need help about 2 constants, digest of 3 messages in primeform Yahoo group, Mar 10, 2006. [Cached copy]
Rick Kreminski, Using Simpson's rule to approximate sums of infinite series, College Math. J. 28 (1997), 368376.
R. J. Mathar, The series limit of sum_k 1/[k*log k *(log log k)^2], arXiv:0902.0789 [math.NA], 2009, App. A.
S.O.S. Math, Bertrand Series.
Eric Weisstein's World of Mathematics, Convergent Series
Wikipédia, Série de Bertrand (in French).


EXAMPLE

2.10974280123689197447925719761655132638553198439474202264991560319281...


MATHEMATICA

digits = 150; NSum[1/(n*Log[n]^2), {n, 2, Infinity}, NSumTerms > 200000, WorkingPrecision > digits + 5, Method > {"EulerMaclaurin", Method > {"NIntegrate", "MaxRecursion" > 20}}] (* Vaclav Kotesovec, Mar 01 2016, after JeanFrançois Alcover *)


CROSSREFS

Cf. A145419, A145420, A145421.
Cf. A137245, A257812. A097906 is a similar sum.
Cf. A347145.
Sequence in context: A158335 A111595 A021478 * A293881 A185285 A268434
Adjacent sequences: A115560 A115561 A115562 * A115564 A115565 A115566


KEYWORD

cons,nonn


AUTHOR

Pierre CAMI, Mar 11 2006


EXTENSIONS

Removed incorrect speculations about relations to A097906  R. J. Mathar, Oct 14 2010
More terms from Robert G. Wilson v, Dec 12 2012
Corrected a(55) and beyond, Vaclav Kotesovec, Mar 01 2016


STATUS

approved



