A115563 Decimal expansion of Sum_{n>=2} 1/(n*log(n)^2).
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
Offset: 1
Examples
2.10974280123689197447925719761655132638553198439474202264991560319281...
Links
- 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).
Crossrefs
Programs
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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 *) maxiter = 20; nn = 10000; alfa = 2; bas = Sum[1/(k*Log[k]^alfa), {k, 2, nn}] + 1/((alfa - 1)*Log[nn + 1/2]^(alfa - 1)); sub = 0; Do[sub = sub + 1/4^s/(2*s + 1)! * NSum[(D[1/(x*Log[x]^alfa), {x, 2 s}]) /. x -> k, {k, nn + 1, Infinity}, WorkingPrecision -> 120, NSumTerms -> 100000, PrecisionGoal -> 120, Method -> {"NIntegrate", "MaxRecursion" -> 100}]; Print[bas - sub], {s, 1, maxiter}] (* Vaclav Kotesovec, Jun 12 2022 *)
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
Comments