Answer
Converges
Work Step by Step
Apply the ratio test to the series
$\Sigma^{\infty} _{n=1} (\frac{n}{2n+1})^n $
$\lim\limits_{n \to \infty} \sqrt[n] {|(\frac{n}{2n+1})^n|}$, Absolute values fall out because all values are positive
$\lim\limits_{n \to \infty} \frac{n}{2n+1} = \frac{1}{2} \lt 1$
Because the answer is less than one, the series converges
