Answer
Radius of convergence is $\sqrt R$
Work Step by Step
Let $x^{2}=y$, then
$\sum_{n=1}^{\infty}c_{n}x^{2n}=\sum_{n=1}^{\infty}c_{n}y^{n}$
So it is convergent for $-R\lt y\lt R$. Therefore, for $x$ it is
convergent if $-\sqrt R\lt x\lt \sqrt R$
Hence, the radius of convergence is $\sqrt R$.