Originally Posted by

**Thomas154321** 1) $\displaystyle Let~M\in\mathbb{R}>0$ Show that $\displaystyle \sum_{n=1}^{\infty} \frac{z^{2n}}{n!}$ converges uniformly for $\displaystyle |z| < M $.

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

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

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