Answer
Converges
Work Step by Step
Given
$$\sum_{n=1}^{\infty}\left(1+\frac{1}{n}\right)^{-n^2}$$
By using the Root Test, we get:
\begin{align*}
\rho&=\lim _{n\rightarrow \infty}\sqrt[n]{a_{n}}\\
&=\lim _{n\rightarrow \infty}\sqrt[n]{ \left(1+\frac{1}{n}\right)^{-n^2}}\\
&= \lim _{n \rightarrow \infty}\left(1+\frac{1}{n}\right)^{-n}\\
&= \frac{1}{e}<1
\end{align*}
Thus the series converges.
