## Calculus (3rd Edition)

$x\lt0$.
We apply the ratio test $$\rho=\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|=\lim _{n \rightarrow \infty} |\frac{ x^{n+1} (n+1)! }{ x^{n} n! }|= |x|\lim _{n \rightarrow \infty} n+1$$ Hence, the series $\Sigma_{n=0}^{\infty} x^{n} n!$ converges if and only if $\rho= |x|\lim _{n \rightarrow \infty} n+1 \lt1$, which is true only for the values $x\lt 0$.