Answer
Converges
Work Step by Step
Here, we have $\lim\limits_{n \to \infty} |\dfrac{a_{n+1}}{a_{n}} |=\lim\limits_{n \to \infty}|\dfrac{\dfrac{(n+1)!}{n^{n+1}}}{\dfrac{n!}{n^n}}|$
$\implies \lim\limits_{n \to \infty}|(\dfrac{n^n (n+1)}{(n+1)^n(n+1)})|=\lim\limits_{n \to \infty}|(1-\dfrac{1}{n+1)})^n|$
This implies that $\lim\limits_{n \to \infty}|\dfrac{(1-\dfrac{1}{n+1)})^{(n+1)}}{(1-\dfrac{1}{n+1)})^n}|=\dfrac{(e^{-1})}{1}=\dfrac{1}{e} \lt 1$
Thus, the series converges by the ratio test.