TITLE: The Sum of Geometric-Factorial Series
QUESTION [5 upvotes]: I was pondering over my Sequences & Series homework, when the following series struck my mind:

$0!+1!x+2!x^{2}+3!x^{3}+.......+n!x^{n}$

And believe me, I am serious that I couldn't think of any approach, but it looked beautiful.I tried to look up all possible summation methods, but it seemed like they didn't offer a break to me.  So guys, any ideas or approach?

REPLY [3 votes]: It is not so bad if you are aware of the incomplete gamma function since
$$f_n(x)=\sum_{k=0}^n k! \,x^k$$
$$f_n(x)=-\frac{e^{-1/x}}{x}\Bigg[(-1)^n (n+1)!\, \Gamma \left(-(n+1),-\frac{1}{x}\right)+\Gamma
   \left(0,-\frac{1}{x}\right) \Bigg]$$ For $x=1$ appears the subfactorial functions and, for $x>1$ the exponential integral function.