Answer
The radius of convergence is $4$.
Work Step by Step
We apply the ratio test
$$
\rho=\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|=\lim _{n \rightarrow \infty} |\frac{ x^{3n+4} /64^{n+1} }{ x^{3n+1} /64^{n} }|= |\frac{ x^{3 } }{ 64}|
$$
Hence, the series $\Sigma_{n=0}^{\infty} x^{3n+1} /64^{n}$ converges if and only if $\rho= |\frac{ x^{3 } }{ 64}| \lt1$. That is, the interval of convergence is $(-4,4)$ and the radius of convergence is $4$.
