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Decimal expansion of Sum_{n>=1} 1/(n*n!).
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%I #44 Feb 13 2024 06:46:42

%S 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,

%T 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,

%U 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

%N Decimal expansion of Sum_{n>=1} 1/(n*n!).

%H G. C. Greubel, <a href="/A229837/b229837.txt">Table of n, a(n) for n = 1..10000</a>

%H Stephen Crowley, <a href="http://arxiv.org/abs/1207.1126">Two New Zeta Constants</a>, arXiv:1207.1126 [math.NT], 2012, page 17.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Logarithmic_integral_function">Logarithmic integral function</a>.

%F Sum_{n >= 1} 1/(n*n!) = Ei(1)-gamma where Ei is the exponential integral and gamma is Euler's constant.

%F Also pFq(1,1; 2,2; 1) where pFq is the generalized hypergeometric function.

%F Also li(e)-gamma, e being the Euler constant (A001113) and li the logarithmic integral function. - _Stanislav Sykora_, May 09 2015

%F 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

%F From _Amiram Eldar_, Aug 01 2020: (Start)

%F Equals Sum_{k>=1} H(k)*k/(k+1)!, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number.

%F Equals Integral_{x=0..1} (exp(x) - 1)/x dx.

%F Equals -Integral_{x=0..1} exp(x)*log(x) dx.

%F Equals -Integral_{x=1..e} log(log(x)) dx. (End)

%F 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

%e 1.3179021514544038948600088442492318379749012457927839928404611969976461...

%p evalf(Ei(1)-gamma,120); # _Vaclav Kotesovec_, May 10 2015

%t RealDigits[ ExpIntegralEi[1] - EulerGamma, 10, 100] // First

%o (PARI) -Euler-real(eint1(-1)) \\ _Charles R Greathouse IV_, Oct 01 2013

%Y Cf. A001008, A001113, A001620, A002805, A091725, A061572, A264806.

%K nonn,cons

%O 1,2

%A _Jean-François Alcover_, Oct 01 2013