TITLE: What is the value of $\sum_{k=1}^{n}k!$?
QUESTION [5 upvotes]: What is the sum of all the factorials starting from 1 to n? Is there any generalized formula for such summation?

REPLY [7 votes]: This is one of the cases where appears the subfactorial function
$$\sum_{k=1}^{n}k!=-1-!1-(-1)^n(n+1)!\times !(-2-n)$$ and,as you will notice in the Wikipedia page, $$!m = \left[ \frac{m!}{e} \right]  = \left\lfloor\frac{m!}{e}+\frac{1}{2}\right\rfloor, \quad m\geq 1$$ or, more generally $$!m=\frac{\Gamma (m+1,-1)}{e}$$ where appears the incomplete gamma function (see here).