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