Answer
Converges
Work Step by Step
Apply the root test to the series
$\Sigma^{\infty} _{n=1} (\frac{n-2}{5n+1})^n $
$\lim\limits_{n \to \infty} \sqrt[n] {|(\frac{n-2}{5n+1})^n|}$, simplify
$\lim\limits_{n \to \infty} \frac{n-2}{5n+1}= \frac{1}{5} \lt 1$
Because the answer is less than 1, the series converges by the root test.
