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%I #52 Jan 01 2024 13:16:33
%S 7,8,8,5,3,0,5,6,5,9,1,1,5,0,8,9,6,1,0,6,0,2,7,6,3,2,3,4,5,4,5,5,4,6,
%T 6,6,4,7,2,7,4,9,6,6,8,2,2,3,2,8,1,6,4,9,7,5,5,1,5,6,4,0,2,3,0,1,7,8,
%U 0,6,4,3,5,6,3,3,0,1,6,2,2,8,7,4,7,1,5,9,2,1,3,3,2,2,4,3,1,9,6,7,5,6
%N Decimal expansion of the number c = Sum_{n>=1} (zeta(n+1)-1)/n.
%C The Alladi-Grinstead constant (A085291) is exp(c-1).
%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="/A085361/b085361.txt">Table of n, a(n) for n = 0..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. 528 and 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, Vol. 29, No. 2 (June 1988), pp. 196-205.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Alladi-GrinsteadConstant.html">Alladi-Grinstead Constant</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ConvergenceImprovement.html">Convergence Improvement</a>.
%F Equals Sum_{n>=2} log(n/(n-1))/n = Sum_{n>=1, k>=2} 1/(n*k^(n+1)). [From Mathworld links]
%F Equals -Sum_{k>=2} (-1)^k * zeta'(k). - _Vaclav Kotesovec_, Jun 17 2021
%F Equals log(A245254) = Sum_{k>=1} log(k)/(k*(k+1)). - _Amiram Eldar_, Jun 27 2021
%F Equals -log(A242624). - _Amiram Eldar_, Feb 06 2022
%e 0.78853056591150896106027632345455466647274966822328164975515640230178...
%p evalf(sum((Zeta(n+1)-1)/n, n=1..infinity), 120); # _Vaclav Kotesovec_, Dec 11 2015
%p evalf(Sum(-(-1)^k*Zeta(1, k), k = 2..infinity), 120); # _Vaclav Kotesovec_, Jun 18 2021
%t Sum[(-1+Zeta[1+n])/n,{n,Infinity}]
%t NSum[Log[k]/(k*(k+1)), {k, 1, Infinity}, WorkingPrecision -> 120, NSumTerms ->5000, Method -> {NIntegrate, MaxRecursion -> 100}] (* _Vaclav Kotesovec_, Dec 11 2015 *)
%o (PARI) suminf(n=1,(zeta(n+1)-1-2^(-n-1))/n)+log(2)/2 \\ _Charles R Greathouse IV_, Feb 20 2012
%o (PARI) sumalt(k=2, -(-1)^k * zeta'(k)) \\ _Vaclav Kotesovec_, Jun 17 2021
%o (Sage)
%o import mpmath
%o mpmath.mp.pretty=True; mpmath.mp.dps=108 #precision
%o mpmath.nsum(lambda n: (-1+mpmath.zeta(1+n))/n, [1,mpmath.inf]) # _Peter Luschny_, Jul 14 2012
%o (Sage) numerical_approx(sum((zeta(k+1)-1)/k for k in [1..1000]), digits=120) # _G. C. Greubel_, Nov 15 2018
%o (Magma) SetDefaultRealField(RealField(120)); L:=RiemannZeta(); (&+[(Evaluate(L,n+1)-1)/n: n in [1..1000]]); // _G. C. Greubel_, Nov 15 2018
%Y Cf. A002210, A085291, A242624, A244109, A245254.
%K nonn,cons
%O 0,1
%A _Eric W. Weisstein_, Jun 25 2003
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