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

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, 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, 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

Offset: 0

Author

Vaclav Kotesovec, Mar 04 2016

Keywords

• nonn
• cons

Examples

			0.659815254349999514863844174352958996077770074088808541384121349320633989...
		

Links

• J. Sondow and E. W. Weisstein, MathWorld: Riemann Zeta Function

Crossrefs

Cf. A093720.

Programs

Maple

evalf(Sum(exp(-1/n)-1+1/n, n=1..infinity), 120);

Mathematica

RealDigits[NSum[Exp[-1/n] - 1 + 1/n, {n, 1, Infinity}, WorkingPrecision -> 200, NSumTerms -> 10000, PrecisionGoal -> 120, Method -> {"NIntegrate", "MaxRecursion" -> 100}]][[1]]

Formula

Equals Sum_{k>=1} (exp(-1/k) - 1 + 1/k).

Comment from Velin Yanev, Mar 03 2023 (Start)

Apparently equals 1/2 - Integral_{x=0..oo} (coth(Pi/x)*(sin(x)/x^2 - 1/x) + 1/Pi) dx.

The proposed expression is difficult to evaluate to arbitrary precision.

Maple code: evalf[50](1/2 - Int(coth(Pi/x)*(sin(x)/x^2 - 1/x) + 1/Pi, x = 0 .. infinity));

Mathematica code: 1/2-NIntegrate[Coth[Pi/t] (Sin[t]/t^2-1/t)+1/Pi,{t,0,Infinity},WorkingPrecision->50,MinRecursion->7]

(End)