Answer
The radius of convergence is $\sqrt 3$.
Work Step by Step
We apply the ratio test:
$$
\rho=\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|=\lim _{n \rightarrow \infty} |\frac{ x^{2n+2} /3^{n+1} }{ x^{2n} /3^{n} }|= x^2/3
$$
Hence, the series $\Sigma_{n=0}^{\infty} x^{2n} /3^{n}$ converges if and only if $\rho= x^2/3| \lt1$. That is, the interval of convergence is $(-\sqrt 3,\sqrt 3)$ and the radius of convergence is $\sqrt 3$.
