Is there any formula for $\sum_{k=1}^n k!$?

Question

Do we have any formula for the sum of factorials above?

There's no nice formula, but there is the formula given here using the $Ei$ and $E_n$ functions — Ben Grossmann, Aug 23 '16 at 14:34
You can find more info on this sequence here → https://oeis.org/A003422 but I guess you will be disappointed (I will post one to grab some points, though, because I'm that shallow). — PseudoNeo, Aug 23 '16 at 14:35

score 2 · Answer 1 · answered Aug 23 '16 at 14:38

2

According to its OEIS page, we have $$\sum_{k=0}^{n-1} k! = \int_0^{+\infty} \frac{x^n - 1}{x - 1}\, e^{-x}\, dx.$$

(I know it's probably not the kind of formulae you're after, but there is not a fundamentally better answer.)

answered Aug 23 '16 at 14:38

PseudoNeo

score 0 · Answer 2 · answered Aug 23 '16 at 14:38

0

There is not a closed 'closed' form, but:

$$\sum_{k=a}^nk!=-(-1)^a\Gamma(a+1)\cdot!(-a-1)-(-1)^n\Gamma(n+2)\cdot!(-n-2)$$

Where $\Gamma(x)$ is the gamma function and $!n$ is the subfactorial function.

answered Aug 23 '16 at 14:38

Jan Eerland

2 Answers2