Sum of series $1 - nx + n(n-1)x^2 - n(n-1)(n-2) x^3+\cdots+ (-1)^n n! x^n$

Question

I am having a hard time summing the seemingly simple finite series:

$$1 - nx + n(n-1)x^2 - n(n-1)(n-2) x^3+\cdots+ (-1)^n n! x^n$$

Thanks for your help in advance!

It isn't quite clear what you're attempting to do, here. It's fine as it stands, so far as I can tell. Are you trying to factor it, or something?
The closed form for $$\sum_{k=0}^n k!\binom{n}{k}(-x)^k$$ involves the incomplete gamma function, which is nonelementary. Are you fine with that?
@Babak: I presume you missed the word finite in the question, then...
@J.M.: Yes, I am looking for the closed form if one exists. What is it in terms of the incomplete Gamma function? Thanks!

Gone · Answer 1 · 2012-06-27 20:20:31Z

Hint $\ $ If you divide by $\rm\:x^n,\:$ differentiate, multiply by $\rm\:x^{n+2},\:$ you'll find that it satisfies the ODE

$$\rm\ x^2 y' - (1+nx) y\, =\, -1$$

which yields the "closed form"

$$\rm\: y = -x^n {\it e}^{-1/x} \int {\it e}^{1/x}x^{-n-2}$$

which is expressible in closed form in terms of the incomplete gamma function - see below.

Or one may use operator methods, or a computer algebra system, e.g. Mma below, with $c_1 = 0$ enter image description here

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