# What is the value of $\sum_{k=1}^{n}k!$? [duplicate]

What is the sum of all the factorials starting from 1 to n? Is there any generalized formula for such summation?

## marked as duplicate by J. M. is a poor mathematician, José Carlos Santos, Henrik, Martin Sleziak, mlcAug 12 '17 at 15:03

• oeis.org/A003422 has some information about the related sum which starts from 0 instead of 1. There is almost certainly not a nice formula (unless you would consider $\sum_1^n k!$ to already be a nice formula). – Micah Sep 22 '16 at 4:44
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).
• +1. Nice answer. Note a 'minor typo': Wi$\color{#f00}{\large\mathrm{l}}$ipedia. – Felix Marin Sep 23 '16 at 4:40