A244109 Decimal expansion of a partial sum limiting constant related to the Lüroth representation of real numbers.
2, 0, 4, 6, 2, 7, 7, 4, 5, 2, 8, 5, 5, 8, 7, 8, 5, 9, 1, 0, 7, 0, 1, 7, 6, 1, 5, 3, 9, 5, 0, 4, 3, 6, 1, 9, 4, 9, 8, 4, 2, 9, 0, 5, 5, 8, 7, 3, 2, 1, 6, 6, 5, 1, 8, 7, 3, 2, 6, 9, 7, 2, 3, 5, 8, 2, 4, 3, 3, 0, 6, 3, 8, 4, 5, 7, 0, 4, 6, 5, 5, 7, 8, 4, 5, 5, 0, 6, 3, 9, 4, 4, 8, 2, 4, 3, 4, 1, 7, 5, 0, 0, 2, 1, 4
Offset: 1
Examples
2.04627745285587859107017615395043619498429055873216651873269723582433...
References
- Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 1.8.1 Alternative representations [of real numbers], p. 62.
Links
- G. C. Greubel, Table of n, a(n) for n = 1..1000
- Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 538.
- Sofia Kalpazidou, Khintchine's constant for Lüroth representation, Journal of Number Theory, Volume 29, Issue 2, June 1988, Pages 196-205.
Programs
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Magma
SetDefaultRealField(RealField(120)); L:=RiemannZeta(); (&+[((1-(-1)^n)*Evaluate(L,n+1)-1)/n: n in [1..1000]]); // G. C. Greubel, Nov 15 2018
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Maple
evalf(Sum(((1 + (-1)^(n+1))*Zeta(n+1) - 1)/n, n=1..infinity), 120); # Vaclav Kotesovec, Dec 11 2015
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Mathematica
NSum[Log[k*(k+1)]/(k*(k+1)), {k, 1, Infinity}, WorkingPrecision -> 120, NSumTerms -> 5000, Method -> {NIntegrate, MaxRecursion -> 100}] (* Vaclav Kotesovec, Dec 11 2015 *) digits = 120; RealDigits[NSum[((1-(-1)^n)*Zeta[n+1] -1)/n, {n, 1, Infinity}, NSumTerms -> 20*digits, WorkingPrecision -> 10*digits, Method -> "AlternatingSigns"], 10, digits][[1]] (* G. C. Greubel, Nov 15 2018 *) -
PARI
default(realprecision, 1000); s = sumalt(n=1, ((1 + (-1)^(n+1))*zeta(n+1) - 1)/n); default(realprecision, 100); print(s) \\ Vaclav Kotesovec, Dec 11 2015
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PARI
2*suminf(k=1, -zeta'(2*k)) \\ Vaclav Kotesovec, Jun 17 2021
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Sage
numerical_approx(sum(((1-(-1)^k)*zeta(k+1)-1)/k for k in [1..1000]), digits=120) # G. C. Greubel, Nov 15 2018
Formula
Equals Sum_{k>=1} log(k*(k+1))/(k*(k+1)).
Equals Sum_{n >=1} ((1 + (-1)^(n+1))*zeta(n + 1) - 1)/n. - G. C. Greubel, Nov 15 2018
Equals 2*Sum_{k>=2} log(k)/(k^2-1) = 2*A340440. - Gleb Koloskov, May 02 2021
Equals -2*Sum_{k>=1} zeta'(2*k). - Vaclav Kotesovec, Jun 17 2021
Extensions
Corrected by Vaclav Kotesovec, Dec 11 2015