Find the radius of convergence for the power series

Question

Find the radius of convergence:

a) $\sum_{n=0}^\infty (-1+(1)^)z^n$

b) $\sum_{n=0}^\infty (1^{-n^2})z^n$

These problems are different from what I encountered before. From what I remember, I use the ratio test. Does this method still apply here?

Answer 1

a) $R=\frac{1}{\lim_{n \rightarrow \infty}|e^{in}|^\frac{1}{n}}$ and $|e^{in}|=|\cos n+i\sin n|=1$

b) $R=\frac{1}{\lim_{n \rightarrow \infty}|e^{-n^2}|^\frac{1}{n}}=\frac{1}{\lim_{n \rightarrow \infty}e^{-n}}=\infty$