1) Let~M\in\mathbb{R}>0 Show that \sum_{n=1}^{\infty} \frac{z^{2n}}{n!} converges uniformly for |z| < M .


That part I feel confident on. I use ratio test on \sum_{n=1}^{\infty} \frac{M^{2n}}{n!} and find convergence, then use Weierstrauss M-Test to show the statement as required.


2) Does \sum_{n=1}^{\infty} \frac{z^{2n}}{n!} converge uniformly on \mathbb{C}?

Here is where I'm a bit confused. Part 1 shows uniform convergence for |z| < M and doesn't set any requirements on M, but the only example I have says the answer is still no, as z=0 is convergence, but not uniformly. A bit of explanation would be wonderful! Thanks