Convergence radius of complex series

Question

I want to find the radius of convergence of the following series $$\sum_{n=1}^{\infty}{\frac{(z+2)^{n-1}}{(n+1)^34^n}}$$ I expanded the series and found that I get the result if I use the series $$\sum_{n=0}^{\infty}{\frac{(z+2)^n}{(n+2)^34^{n+1}}}$$ Then, with the root test, I got $$R=\lim_{n\rightarrow{\infty}}\frac{1}{(a_n)^{1/n}}=\lim_{n\rightarrow{\infty}}{(n+2)^{3/n}4^{1+1/n}}=4$$ Is this correct, or is it wrong if I rewrite the first series as the second one?