The sum of reciprocal multifactorials can be given in closed form by the beautiful formula
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  | ![(e^(1/n))/n[n+sum_(k=1)^(n-1)n^(k/n)gamma(k/n,1/n)],](/images/equations/ReciprocalMultifactorialConstant/Inline9.svg) |
(3)
|
where 
is a lower incomplete
gamma function (E. W. Weisstein, Aug. 6, 2008).
This gives the special cases
 |  | ![sqrt(e)[1+sqrt(pi/2)erf(1/2sqrt(2))]](/images/equations/ReciprocalMultifactorialConstant/Inline13.svg) |
(4)
|
 |  | ![1/3e^(1/3)[3+3^(1/3)gamma(1/3,1/3)+3^(2/3)gamma(2/3,1/3)].](/images/equations/ReciprocalMultifactorialConstant/Inline16.svg) |
(5)
|
Reciprocal Multifactorial Constant
See also
MultifactorialExplore with Wolfram|Alpha
Cite this as:
Weisstein, Eric W. "Reciprocal Multifactorial Constant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ReciprocalMultifactorialConstant.html