Answer
Converges
Work Step by Step
Given
$$\sum_{n=1}^{\infty} \frac{10^n}{2^{n^2} }$$
By using the Ratio Test, we get:
\begin{align*}
\rho&=\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|\\
&=\lim _{n \rightarrow \infty} \frac{10^{n+1}}{2^{n^2+2n+1}} \frac{2^{n^2}}{10^n} \\
&= \lim _{n \rightarrow \infty} \frac{10}{2^{2n+1}}\\
&= 0<1
\end{align*}
Thus the series converges.
