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A244109
Decimal expansion of a partial sum limiting constant related to the Lüroth representation of real numbers.
5
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
1,1
REFERENCES
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 1.8.1 Alternative representations [of real numbers], p. 62.
LINKS
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 A085361 + A131688. - Vaclav Kotesovec, Dec 11 2015
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
Cf. A002210, A085361. Equals twice A340440.
Sequence in context: A233673 A319931 A192133 * A133144 A342384 A192134
KEYWORD
nonn,cons
AUTHOR
EXTENSIONS
Corrected by Vaclav Kotesovec, Dec 11 2015
STATUS
approved