Answer
Diverges
Work Step by Step
Apply the ratio test to the series
$\Sigma^{\infty}_{n=1} (\frac{2n}{n+1})^n $
$\lim\limits_{n \to \infty} \sqrt[n] {|(\frac{2n}{n+1})^n|} $
$\lim\limits_{n \to \infty} \frac{2n}{n+1} = 2 \gt 1 $
Because the answer is greater than 1, the series diverges.
