Answer
$\sqrt {17}$
Work Step by Step
The radius of convergence of a power series represented by $\dfrac{A(x)}{B(x)}$ can be defined as the distance from $min(|x_0-y|)$, where y is root of $B(x)$, that is, $B(y)=0$.
We have: $\dfrac{A(x)}{B(x)}=\dfrac{2x}{x^2+16}; x_0=1$
This implies that the root of $B(x)=x^2+16=0 \implies x =\pm 4i$
So, the radius of convergence is $R=|1 \pm 4i|=\sqrt {(1)^2+(4)^2}=\sqrt {17}$
