Answer
Convergent
Work Step by Step
Use the Root Test: $\lim\limits_{n\to\infty}\sqrt[n]{|a_n|}$
$$\lim\limits_{n\to\infty}\sqrt[n]{|a_n|}=\lim\limits_{n\to\infty}\sqrt[n]{\left(\frac{n}{n+1}\right)^{n^2}}=\lim\limits_{n\to\infty}\left(\frac{n}{n+1}\right)^{\frac{n^2}{n}} =\lim\limits_{n\to\infty}\left(\frac{n}{n+1}\right)^{n}$$
Divide the top and bottom by the highest power of $n$.
$$\lim\limits_{n\to\infty}\left(\frac{1}{1+\frac{1}{n}}\right)^{n}=\frac{1}{\lim\limits_{n\to\infty}(1+\frac{1}{n})^n}$$
We know that $\lim\limits_{n\to \infty}(1+\frac{1}{n})^n$ is the definition of $e$.
Therefore, $$\frac{1}{\lim\limits_{n\to\infty}(1+\frac{1}{n})^n}=\frac{1}{e}<1$$
The series is convergent by the Root Test.