Answer
Diverges
Work Step by Step
Given
$$\sum_{n=1}^{\infty} \frac{2^{n^{2}}}{n !}$$
By using the Ratio test
\begin{align*}
\rho&=\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|\\
&=\lim _{n \rightarrow \infty} \frac{2^{n^{2}+2n+1}}{(n+1) !} \frac{n !}{2^{n^{2}}}\\
&=\lim _{n \rightarrow \infty} \frac{2^{ 2n+1}}{(n+1) } \\
&=\infty
\end{align*}
Thus, the given series diverges.
