Answer
Divergent
Work Step by Step
By the Ratio Test, we have
$\lim_{n\to\infty}\biggl|\frac{e^{n+1}}{(n+1)^2}\frac{n^2}{e^n}\biggr|=\lim_{n\to\infty}\biggl|\frac{n^2e}{n^2+2n+1}\biggr|=e > 1$
Since the limit $ > 1$, the series is divergent by the Ratio Test