Answer
Divergent
Work Step by Step
$\sum_{n=2}^{\infty}a_{n}=\sum_{n=2}^{\infty}(\dfrac{-2n}{n+1})^{5n}$
$|a_{n}|=(\frac{2n}{n+1})^{5n}$
$\lim\limits_{n \to \infty} \sqrt[n] |a_{n}|=\lim\limits_{n \to \infty} \sqrt[n] {(\frac{2n}{n+1})^{5n}}$
$=\lim\limits_{n \to \infty} {(\frac{2n}{n+1})^{5}}$
$=\lim\limits_{n \to \infty} {(\frac{2n}{n})^{5}}$
$=2^{5}$
$=32 \gt 1$
The series is divergent.