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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 (list; constant; graph; refs; listen; history; text; internal format)
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), 302-313.

Bart Braden, Calculating sums of infinite series, Am. Math. Monthly 99 (1992) 649-655.

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), 368-376.

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 Jean-François 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

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Last modified December 7 12:17 EST 2021. Contains 349581 sequences. (Running on oeis4.)