# The Sum of Geometric-Factorial Series

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?

• There is an awful closed form for $x=1$ (With an added $1$) mathworld.wolfram.com/FactorialSums.html But I don’t think it will have a nice form in general Commented Dec 2, 2020 at 7:09
• Is the first term meant to be $0!$ instead of $0!x$?
– J.G.
Commented Dec 2, 2020 at 8:23
• The suggested polynomial is a type of Left Factorial polynomial. Left Factorials are fun to learn about and rarely spotted in the wild. An example of some classes of Left Factorial polynomials may be found in the paper Generalized Factorial Functions, Numbers and Polynomials Commented Dec 2, 2020 at 8:28
• @J.G. Thanks for noticing, edited :) Commented Dec 2, 2020 at 12:05
• @Leucippus Thanks for the pointer, will check 'em out... Commented Dec 2, 2020 at 12:07

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.