Answer
The series converges by the root test.
Work Step by Step
Use the Root test: $\lim\limits_{n \to \infty}\sqrt[n] {|a_n|}$
Take the nth root of the series and then take the limit as $n\to \infty$ $\lim\limits_{n \to \infty}\sqrt[n]{\left|\frac{(2n+1)^n}{n^{2n}}\right|}=\lim\limits_{n \to \infty}\frac{2n+1}{n^2}$
Split the fraction and simplify.
$\lim\limits_{n \to \infty}\left(\frac{2n}{n^2}+\frac{1}{n^2}\right)=\lim\limits_{n \to \infty}\left(\frac{2}{n}+\frac{1}{n^2}\right)$
So as $n\to\infty$, both terms approach zero, thus,
$\lim\limits_{n \to \infty}\left(\frac{2}{n}+\frac{1}{n^2}\right)=0<1$
Therefore, the series $\sum_{n=1}^{\infty}\frac{(2n+1)^n}{n^{2n}}$ converges by the Root Test.
