Find the region of convergence of the series

Question

Please tell me how to find the region of convergence of following series

$$\sum_{n=1}^\infty \frac{(-1)^{n-1} z^{2n-1}}{(2n-1)!}$$

I applied the ratio test for this and got

$$\lim_{n \to \infty} \left|\frac{-z^2}{(2n+1)(2n)}\right|= 0 $$ After this ...what should I do?

Answer 1

Since that limit is indeed $0$, for whatever value of $z$, the series converges always. That is, the radius of convergence is $+\infty$.