## Calculus 8th Edition

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.