Answer
The series converges for all $x$ and the radius of converges is $R=\infty$
Work Step by Step
Apply the ratio test:
\begin{aligned}
\rho &=\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|\\
&=\lim _{n \rightarrow \infty}\left|\frac{(2(n+1)) ! x^{n+1}}{((n+1) !)^{3}} \cdot \frac{(n !)^{3}}{(2 n) ! x^{n}}\right|\\
&=\lim _{n \rightarrow \infty}\left|x \frac{(2 n+2)(2 n+1)}{(n+1)^{3}}\right| \\
&=\lim _{n \rightarrow \infty}\left|x \frac{4 n^{2}+6 n+2}{n^{3}+3 n^{2}+3 n+1}\right|\\
&=\lim _{n \rightarrow \infty}\left|x \frac{4 n^{-1}+6 n^{-1}+2 n^{-3}}{1+3 n^{-1}+3 n^{-2}+n^{-3}}\right|=0
\end{aligned}
Then the series converges for all $x$ and the radius of converges is $R=\infty$
