TITLE: Compact formula for $\sum_k k!$
QUESTION [7 upvotes]: Is there any compact formula for:
$$\sum_{k=0}^n k!$$
I've tried to find it using one method for summation, but I was able to receive only compact formula for $\sum_k k! \cdot k = (n+1)!-1$ 
I've typed it into wolfram, but answer is also pretty complicated.

REPLY [6 votes]: This is A003422; the only more or less closed form expression given there is
$$\sum_{k=0}^{n-1}k!=\int_0^\infty\frac{x^n-1}{x-1}e^{-x}dx\;.$$

REPLY [2 votes]: \begin{align}
\sum_{k=0}^n (k^2+1)k!  &=  \sum_{k=0}^n [(k+1)^2-2k]k!  \\  &= 
 \sum_{k=0}^n (k+1)(k+1)!  -\sum_{k=0}^n 2k \cdot k!  \\  &=  \bigl((n+2)!+1\bigr) -2 \bigl((n+1)!+1\bigr)  \\  &=  n(n+1)! -1
\end{align}