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%I #24 Feb 23 2024 02:16:34
%S 1,0,0,8,3,3,3,6,0,8,9,0,7,2,9,0,2,8,9,9,7,6,4,5,3,6,6,7,3,5,4,8,3,8,
%T 7,8,6,0,7,1,0,7,7,2,8,1,5,7,9,5,4,3,1,0,2,0,0,3,0,5,9,0,7,4,9,2,7,0,
%U 7,5,5,0,4,8,4,8,1,1,1,0,8,4,1,1,4,8,5,5,9,4,1,6,1,7,0,0,6,5,7,8,1,9,2,5,2,6,8,9,9,1,9,4,6,9,7,5,7,7,4,2
%N Decimal expansion of Sum_{k>=0} 1/(5k)!.
%C From _Vaclav Kotesovec_, Feb 24 2016: (Start)
%C Sum_{k>=0} 1/k! = A001113 = exp(1).
%C Sum_{k>=0} 1/(2k)! = A073743 = cosh(1).
%C Sum_{k>=0} 1/(3k)! = A143819 = (2*cos(sqrt(3)/2)*exp(-1/2) + exp(1))/3.
%C Sum_{k>=0} 1/(4k)! = (cos(1) + cosh(1))/2 = 1.0416914703416917479394211141...
%C (End)
%C For q integer >= 1, Sum_{m>=0} 1/(q*m)! = (1/q) * Sum_{k=1..q} exp(X_k) where X_1, X_2, ..., X_q are the q-th roots of unity. - _Bernard Schott_, Mar 02 2020
%C Continued fraction: 1 + 1/(120 - 120/(30241 - 30240/(360361 - ... - P(n-1)/((P(n) + 1) - ... )))), where P(n) = (5*n)*(5*n - 1)*(5*n - 2)*(5*n - 3)*(5*n - 4) for n >= 1. Cf. A346441. - _Peter Bala_, Feb 22 2024
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FactorialSums.html">Factorial Sums</a>
%F Equals Sum_{k>=0} 1/A100734(k).
%F Equals (exp(1) + exp(-(-1)^(1/5)) + exp((-1)^(2/5)) + exp(-(-1)^(3/5)) + exp((-1)^(4/5)))/5.
%F Equals (exp(1) + 2*exp(-(sqrt(5) + 1)/4) * cos(sqrt((5 - sqrt(5))/2)/2) + 2*exp((sqrt(5) - 1)/4) * cos(sqrt((5 + sqrt(5))/2)/2))/5. - _Vaclav Kotesovec_, Feb 24 2016
%F Sum_{k>=0} (-1)^k / (5*k)! = (exp(-1) + 2*cos(5^(1/4)/(2*sqrt(phi))) * exp(phi/2) + 2*cos(5^(1/4)*sqrt(phi)/2) / exp(1/(2*phi)))/5 = 0.99166694223909419..., where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - _Vaclav Kotesovec_, Mar 02 2020
%e 1 + 1/5! + 1/10! + 1/15! + ... = 1.008333608907290289976453667354838786...
%p evalf((exp(1) + 2*exp(-(sqrt(5) + 1)/4) * cos(sqrt((5 - sqrt(5))/2)/2) + 2*exp((sqrt(5) - 1)/4) * cos(sqrt((5 + sqrt(5))/2)/2))/5, 120); # _Vaclav Kotesovec_, Feb 24 2016
%t RealDigits[HypergeometricPFQ[{}, {1/5, 2/5, 3/5, 4/5}, 1/3125], 10, 120][[1]]
%o (PARI) suminf(k=0, 1/(5*k)!) \\ _Michel Marcus_, Feb 21 2016
%Y Cf. A100734.
%Y Cf. A001113 (Sum 1/k!), A073743 (Sum 1/(2k)!), A143819 (Sum 1/(3k)!), A332890 (Sum 1/(4k)!), this sequence (Sum 1/(5k)!), A332892 (Sum 1/(6k)!), A346441.
%K nonn,cons
%O 1,4
%A _Ilya Gutkovskiy_, Feb 21 2016
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