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This question already has an answer here:

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

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marked as duplicate by Omnomnomnom, Namaste algebra-precalculus Aug 23 '16 at 14:42

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    $\begingroup$ There's no nice formula, but there is the formula given here using the $Ei$ and $E_n$ functions $\endgroup$ – Omnomnomnom Aug 23 '16 at 14:34
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    $\begingroup$ You can find more info on this sequence here → oeis.org/A003422 but I guess you will be disappointed (I will post one to grab some points, though, because I'm that shallow). $\endgroup$ – PseudoNeo Aug 23 '16 at 14:35
  • $\begingroup$ Nice, quick find, @Omnomnomnom ! $\endgroup$ – Namaste Aug 23 '16 at 14:44
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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.)

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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.

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