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