I have 2 questions with one of them being a bit confusing the first one is prove that the series $$\sum_{n=1}^\infty\cos(z/n^5)/n!$$ converges uniformly on D(0,1)
And the second question this is all the info I have
Calculate $$\sum_{n=0}^\infty z^n/2^n$$
For this one I think I have to use contour integrals $$\int_{\gamma } f(z)dz=\sum_{n=0}^{\infty} z^{n}dz$$
but im not sure what to do with that
You should have posted two separate questions. Anyway, the answer to the second question is $\dfrac1{1-\frac z2}$, because, if $|z|<2$, then$$\sum_{n=0}^\infty\frac{z^n}{2^n}=\sum_{n=0}^\infty\left(\frac z2\right)^n=\frac1{1-\frac z2}.$$
The series of the first question converges uniformly in $D(0,1)$ indeed. Let $M=\sup_{|z|\leqslant 1}\bigl|\cos(z)\bigr|$. Then$$(\forall z\in D(0,1))(\forall n\in\mathbb{N}):\bigl|\cos(z)\bigr|\leqslant M$$and therefore you can use the fact that the series $\sum_{n=1}^\infty\frac M{n!}$ converges to deduce, from the $M$-test, that your series converges uniformly on $D(0,1)$.