A244109 - OEIS

%I #35 Sep 08 2022 08:46:08

%S 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,

%T 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,

%U 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

%N Decimal expansion of a partial sum limiting constant related to the Lüroth representation of real numbers.

%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 1.8.1 Alternative representations [of real numbers], p. 62.

%H G. C. Greubel, <a href="/A244109/b244109.txt">Table of n, a(n) for n = 1..1000</a>

%H Steven R. Finch, <a href="https://doi.org/10.1017/9781316997741">Mathematical Constants II</a>, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 538.

%H Sofia Kalpazidou, <a href="http://dx.doi.org/10.1016/0022-314X(88)90099-6">Khintchine's constant for Lüroth representation</a>, Journal of Number Theory, Volume 29, Issue 2, June 1988, Pages 196-205.

%F Equals Sum_{k>=1} log(k*(k+1))/(k*(k+1)).

%F Equals A085361 + A131688. - _Vaclav Kotesovec_, Dec 11 2015

%F Equals Sum_{n >=1} ((1 + (-1)^(n+1))*zeta(n + 1) - 1)/n. - _G. C. Greubel_, Nov 15 2018

%F Equals 2*Sum_{k>=2} log(k)/(k^2-1) = 2*A340440. - _Gleb Koloskov_, May 02 2021

%F Equals -2*Sum_{k>=1} zeta'(2*k). - _Vaclav Kotesovec_, Jun 17 2021

%e 2.04627745285587859107017615395043619498429055873216651873269723582433...

%p evalf(Sum(((1 + (-1)^(n+1))*Zeta(n+1) - 1)/n, n=1..infinity), 120); # _Vaclav Kotesovec_, Dec 11 2015

%t NSum[Log[k*(k+1)]/(k*(k+1)), {k, 1, Infinity}, WorkingPrecision -> 120, NSumTerms -> 5000, Method -> {NIntegrate, MaxRecursion -> 100}] (* _Vaclav Kotesovec_, Dec 11 2015 *)

%t 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 *)

%o (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

%o (PARI) 2*suminf(k=1, -zeta'(2*k)) \\ _Vaclav Kotesovec_, Jun 17 2021

%o (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

%o (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

%Y Cf. A002210, A085361. Equals twice A340440.

%K nonn,cons

%O 1,1

%A _Jean-François Alcover_, Jun 20 2014

%E Corrected by _Vaclav Kotesovec_, Dec 11 2015