For part 2), remember the expansion for .
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
Just use the ratio test for part 2.
The series is convergent wherever $\displaystyle \begin{align*} \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| < 1 \end{align*}$, so solve
$\displaystyle \begin{align*} \lim_{n \to \infty} \left| \frac{\frac{z^{2 \left( n + 1 \right) } }{\left( n + 1 \right) !}}{\frac{z^{2n}}{n!}} \right| &< 1 \\ \lim_{n \to \infty} \left| \frac{n!\,z^{2n + 2}}{\left( n + 1 \right) n! \, z^{2n}} \right| &< 1 \\ \lim_{n \to \infty} \left| \frac{z^2}{n + 1} \right| &< 1 \\ \left| z^2 \right| \lim_{n \to \infty} \frac{1}{n + 1} &< 1 \\ \left| z \right| ^2 \cdot 0 &< 1 \\ 0 &< 1 \end{align*}$
The limit of the ratios is 0, which is ALWAYS less than 1, no matter what value of z. So that means that the series is convergent for all z.