1) Show that converges uniformly for .

That part I feel confident on. I use ratio test on and find convergence, then use Weierstrauss M-Test to show the statement as required.

2) Does converge uniformly on ?

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