OFFSET
1,1
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.
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
EXAMPLE
2.04627745285587859107017615395043619498429055873216651873269723582433...
MAPLE
evalf(Sum(((1 + (-1)^(n+1))*Zeta(n+1) - 1)/n, n=1..infinity), 120); # Vaclav Kotesovec, Dec 11 2015
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 *)
PROG
(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
(PARI) 2*suminf(k=1, -zeta'(2*k)) \\ Vaclav Kotesovec, Jun 17 2021
(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
(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
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Jean-François Alcover, Jun 20 2014
EXTENSIONS
Corrected by Vaclav Kotesovec, Dec 11 2015
STATUS
approved