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

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 $c_1 = 0$

