cp's OEIS Frontend

A269768 Decimal expansion of Sum_{n>=2} (-1)^n * zeta(n)/n!.

%I A269768 #14 Feb 16 2025 08:33:30
%S A269768 6,5,9,8,1,5,2,5,4,3,4,9,9,9,9,5,1,4,8,6,3,8,4,4,1,7,4,3,5,2,9,5,8,9,
%T A269768 9,6,0,7,7,7,7,0,0,7,4,0,8,8,8,0,8,5,4,1,3,8,4,1,2,1,3,4,9,3,2,0,6,3,
%U A269768 3,9,8,9,0,7,5,7,3,1,6,7,8,5,1,8,5,7,6,2,4,8,3,0,0,8,7,8,6,0,9,6,0,7,5,8,0,9
%N A269768 Decimal expansion of Sum_{n>=2} (-1)^n * zeta(n)/n!.
%H A269768 J. Sondow and E. W. Weisstein, <a href="https://mathworld.wolfram.com/RiemannZetaFunction.html">MathWorld: Riemann Zeta Function</a>
%F A269768 Equals Sum_{k>=1} (exp(-1/k) - 1 + 1/k).
%F A269768 Comment from _Velin Yanev_, Mar 03 2023 (Start)
%F A269768 Apparently equals 1/2 - Integral_{x=0..oo} (coth(Pi/x)*(sin(x)/x^2 - 1/x) + 1/Pi) dx.
%F A269768 The proposed expression is difficult to evaluate to arbitrary precision.
%F A269768 Maple code: evalf[50](1/2 - Int(coth(Pi/x)*(sin(x)/x^2 - 1/x) + 1/Pi, x = 0 .. infinity));
%F A269768  Mathematica code: 1/2-NIntegrate[Coth[Pi/t] (Sin[t]/t^2-1/t)+1/Pi,{t,0,Infinity},WorkingPrecision->50,MinRecursion->7]
%F A269768 (End)
%e A269768 0.659815254349999514863844174352958996077770074088808541384121349320633989...
%p A269768 evalf(Sum(exp(-1/n)-1+1/n, n=1..infinity), 120);
%t A269768 RealDigits[NSum[Exp[-1/n] - 1 + 1/n, {n, 1, Infinity}, WorkingPrecision -> 200, NSumTerms -> 10000, PrecisionGoal -> 120, Method -> {"NIntegrate", "MaxRecursion" -> 100}]][[1]]
%Y A269768 Cf. A093720.
%K A269768 nonn,cons
%O A269768 0,1
%A A269768 _Vaclav Kotesovec_, Mar 04 2016