## Factorial Sum

Sums with unity Numerator and Factorials in the Denominator which can be expressed analytically include

 (1) (2) (3) (4)

where is a Hypergeometric Function and is a Gamma Function.

Sums with in the Numerator having analytic solutions include

 (5) (6) (7) (8)

A sum with in the Numerator is

 (9)
where is the Hypergeometric Function.

Sums of factorial Powers include

 (10)

 (11)

where
 (12) (13) (14)

(Beeler et al. 1972, Item 116).

References

Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, Feb. 1972.