Answer
Diverges
Work Step by Step
Given
$$\sum_{n=1}^{\infty} \frac{2^n}{n^{100} }$$
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{2^{n+1}}{(n+1)^{100} }\frac{n^{100}}{2^n} \\
&=\lim _{n \rightarrow \infty} \frac{2n^{100}}{(n+1)^{100} }\\
&= 2>1
\end{align*}
Thus the series diverges.
