What is the radius of convergence for $\sum_{n=1}^{\infty}(\sqrt[n]{n}+2)^{n}z^{n}$?

Question

This is what I tried:

Let $a_{n}=(\sqrt[n]{n}+2)^{n}$. Then $\sqrt[n]{a_{n}}=\sqrt[n]{n}+2$. Taking the limit of this as $n\to \infty$ gives 3. Does that mean the radius of convergence is $\frac{1}{3}$?

I feel like I am doing something wrong here.