A229837 Decimal expansion of Sum_{n>=1} 1/(n*n!).
1, 3, 1, 7, 9, 0, 2, 1, 5, 1, 4, 5, 4, 4, 0, 3, 8, 9, 4, 8, 6, 0, 0, 0, 8, 8, 4, 4, 2, 4, 9, 2, 3, 1, 8, 3, 7, 9, 7, 4, 9, 0, 1, 2, 4, 5, 7, 9, 2, 7, 8, 3, 9, 9, 2, 8, 4, 0, 4, 6, 1, 1, 9, 6, 9, 9, 7, 6, 4, 6, 1, 0, 7, 7, 5, 6, 1, 3, 9, 4, 8, 2, 6, 1, 1, 9, 5, 3, 6, 4, 6, 8, 3, 4, 3, 9, 2, 2, 0, 7
Offset: 1
Examples
1.3179021514544038948600088442492318379749012457927839928404611969976461...
Links
- G. C. Greubel, Table of n, a(n) for n = 1..10000
- Stephen Crowley, Two New Zeta Constants: Fractal String, Continued Fraction, and Hypergeometric Aspects of the Riemann Zeta Function, arXiv:1207.1126 [math.NT], 2012, page 17.
- Wikipedia, Logarithmic integral function.
Programs
-
Maple
evalf(Ei(1)-gamma,120); # Vaclav Kotesovec, May 10 2015
-
Mathematica
RealDigits[ ExpIntegralEi[1] - EulerGamma, 10, 100] // First
-
PARI
-Euler-real(eint1(-1)) \\ Charles R Greathouse IV, Oct 01 2013
Formula
Sum_{n >= 1} 1/(n*n!) = Ei(1)-gamma where Ei is the exponential integral and gamma is Euler's constant.
Also pFq(1,1; 2,2; 1) where pFq is the generalized hypergeometric function.
Also li(e)-gamma, e being the Euler constant (A001113) and li the logarithmic integral function. - Stanislav Sykora, May 09 2015
Continued fraction expansion: Ei(1) - gamma = 1/(1 - 1^3/(5 - 2^3/(11 -...-(n-1)^3/(n^2+n-1) -...))). See A061572. - Peter Bala, Feb 01 2017
From Amiram Eldar, Aug 01 2020: (Start)
Equals Integral_{x=0..1} (exp(x) - 1)/x dx.
Equals -Integral_{x=0..1} exp(x)*log(x) dx.
Equals -Integral_{x=1..e} log(log(x)) dx. (End)
Equals e * Sum_{k>=1} (-1)^(k+1)*H(k)/k!, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number. - Amiram Eldar, Jun 25 2021