find radius of convergence of sum(upper bound infinity and lower bound n=0) n!x^n
by the ratio test, the radius of convergence will be given by the bounds of x such that
$\displaystyle \lim_{n \to \infty} \bigg| \frac {(n + 1)! x^{n + 1}}{n!x^n} \bigg| < 1$
be sure to check the end points. see post #2 here on how to do that
you may also want to check this out