Answer
$$\rho = \frac 12$$
Work Step by Step
1. Use the ratio test to test for convergence:
$$\lim_{n \longrightarrow \infty} \Bigg| \frac{ 2^{n+1}x^{n+1} }{2^nx^n} \Bigg | = \lim_{n \longrightarrow \infty} \Bigg| 2x \Bigg| = 2x$$
- Therefore, the series converges absolutely when $2x \lt 1$, or $x \lt \frac{1}{2}$
$$\rho = \frac 12$$