Radius of convergence of $\displaystyle{\sum_{n=0}^{\infty}} {(n!)^3 \over (3n)!}z^{3n} $. 
Find the radius of convergence of $\displaystyle{\sum_{n=0}^{\infty}} {(n!)^3 \over (3n)!}z^{3n} \ ?$

I applied Cauchy-Hadamard test and the result is coming $0$ (radius of convergence). To obtain the limit I also used Cauchy's first limit theorem. For a lot of messy calculation I didn't provide my work.
Please someone check whether I'm right or wrong.
Thank you..

If we represent the factorials everywhere as gamma-functions and Pochhammer symbols, this is actually a generalized hypergeometric function 4F3(...;;;z/3). According to wikipedia the radius of convergence is where the argument becomes 1, which means in our case, |z|=3.
