Answer
$\rho =2$
Work Step by Step
$\sum_{\infty }^{n=0}(\frac{n}{2^n}x^n)$
$\;\;\;\;\;\;\;\;\;\;\;\;\;\rightarrow \lim_{n\rightarrow \infty} \left | \frac{(x^{n+1}(n+1)}{2^{n+1}}\;.\;\frac{2^n}{nx^n} \right |
= \left | \frac{x}{2} \right | \lim_{n\rightarrow \infty} \left | \frac{n+1}{n} \right |$
$|\frac{x}{2}|<1\;\;\;\;\;\;\;\;\;$ $\;\;\;\;\;\;\;\;\;|x|<2$
Diameter of convergence = 4
Radius of convergence $(\rho ) = \frac{diameter}{2} = \frac{4}{2}=2$
$\rho =2$
