OFFSET
1,4
COMMENTS
From Vaclav Kotesovec, Feb 24 2016: (Start)
Sum_{k>=0} 1/k! = A001113 = exp(1).
Sum_{k>=0} 1/(2k)! = A073743 = cosh(1).
Sum_{k>=0} 1/(3k)! = A143819 = (2*cos(sqrt(3)/2)*exp(-1/2) + exp(1))/3.
Sum_{k>=0} 1/(4k)! = (cos(1) + cosh(1))/2 = 1.0416914703416917479394211141...
(End)
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
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
LINKS
Eric Weisstein's World of Mathematics, Factorial Sums
FORMULA
Equals Sum_{k>=0} 1/A100734(k).
Equals (exp(1) + exp(-(-1)^(1/5)) + exp((-1)^(2/5)) + exp(-(-1)^(3/5)) + exp((-1)^(4/5)))/5.
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
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
EXAMPLE
1 + 1/5! + 1/10! + 1/15! + ... = 1.008333608907290289976453667354838786...
MAPLE
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
MATHEMATICA
RealDigits[HypergeometricPFQ[{}, {1/5, 2/5, 3/5, 4/5}, 1/3125], 10, 120][[1]]
PROG
(PARI) suminf(k=0, 1/(5*k)!) \\ Michel Marcus, Feb 21 2016
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Ilya Gutkovskiy, Feb 21 2016
STATUS
approved