Answer
The series converges for all $x$ and 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| \\
&=\left|\frac{\frac{4 \times 4^{n} \times x^{-2} x^{2 n-1}}{(2 n+3)(2 n+2)(2 n+1) !}}{\frac{4^{n} x^{2 n-1}}{(2 n+1) !}}\right| \\
&=\lim _{n \rightarrow \infty}\left|\frac{4 \times 4^{n} \times x^{2 n-1}}{x^{2} \times 4^{n} \cdot x^{2 n-1}} \cdot \frac{(2 n+1) !}{(2 n+3)(2 n+2)(2 n+1) !}\right| \\
&=4 \frac{1}{x^{2}} \lim _{n \rightarrow \infty} \frac{1}{(2 n+3)(2 n+3)}\\
&=0
\end{aligned}
Then the series converges for all $x$ and the radius of converges is $R=\infty$.
