Finding radius of convergence $\sum _{n=0}^{}(2+(-1)^n)^nz^n$

Question

1) $\sum _{n=0}^{}(2+(-1)^n)^nz^n$

2) $\sum _{n=0}^{}n!z^n$

3) $\sum _{n=1}^{}\frac{1}{n^2}(\sqrt{ n^2+n}-\sqrt{n^2+1})^nz^n$

Hello, I struggle to show the radius of convergence for the above functions. I began with the second one and came to the conclusion that the radius of convergence is $0$ but I'm not sure if that's correct. I would appreciate it if you could help me with the other ones!

Answer 1

(2) and (3) left to you. first one is a little bit tricky. $\frac{1}{ROC}= \limsup_{n\to \infty}|a_n|^{\frac{1}{n}}$. $$\limsup_{n\to \infty}|a_n|^{\frac{1}{n}}=3.$$