calculus - Radius of Convergence of $sum_{n = 0}^{infty} frac{(-1)^nn!x^n}{n^n}$

I'm taking the AP Calculus BC Exam next week and ran into this problem with no idea how to solve it. Unfortunately, the answer key didn't provide explanations, and I'd really, really appreciate it if someone could explain why the answer is $e$.

which of the following is the radius of the convergence of the series $\large{\sum\limits_{n=0}^\infty\frac{(-1)^n n!x^n}{n^n}}$

$(A) \quad 0,\qquad (B) \quad\frac1e,\qquad (C)\quad1,\qquad (D) \quad e,\qquad (E) \infty$

Thank you all so much.

Answer

Denote $\sum a_n x^n$ the given series then

$$\left\vert\frac{a_{n+1}}{a_n}\right\vert=\left(1+\frac1n\right)^{-n}\xrightarrow{n\to\infty}\frac1e$$
so by the ratio test, the radius of convergence is $e$.