Answer
Convergent
Work Step by Step
Apply 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 $n$.
$\lim\limits_{n\to\infty}\left(\frac{1}{1+\frac{1}{n}}\right)^{n}=\frac{1}{\lim\limits_{n\to\infty}(1+\frac{1}{n})^n}$
Since, $e=\lim\limits_{n\to \infty}(1+\frac{1}{n})^n$
Therefore,
$\frac{1}{\lim\limits_{n\to\infty}(1+\frac{1}{n})^n}=\frac{1}{e}<1$
Hence, the series is convergent by the Root Test.
