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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!

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    @J.M.: Yes, I am looking for the closed form if one exists. What is it in terms of the incomplete Gamma function? Thanks!2012-06-27

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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