Answer
The series converges when $ |x|\lt 1/2$
Work Step by Step
Given
$$\sum_{n=1}^{\infty} 2^nx^{n}$$
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}\left| \frac{2^{n+1}x^{n+1}}{ 2^nx^{n}}\right|\\
&= \lim _{n \rightarrow \infty} |2x|\\
&= |2x|
\end{align*}
Thus the series converges when $|2x|\lt1\ \to |x|\lt 1/2$.
