Going through some of the assignment sheets before my Complex Analysis exam, and going through the lecture notes I am still unsure how to answer this question.
Conclude the following series converge uniformly on the following domains:
$$\sum_{k=1}^{\infty} \sqrt{k} e^{-kz} \quad \text{on}\quad \{z \in \mathbb{C} : r< \Re(z) \} \quad \text{for any} \quad r>0$$
$$\sum_{k=1}^{\infty} \frac{2^k}{z^k+z^{-k}} \quad \text{on} \quad \{z \in \mathbb{C} : |z| \leq r \} \quad \text{for any} \quad0<r<\frac{1}{2}$$
I assume I have to find an upper bound of the modulus of each terms and show that the sum of these bounds converges, but I am unsure on how to construct these bounds, any guidance would be appreciated thanks!
#1 is $\sum_{k=1}^\infty \sqrt{k} s^k$ where $s = e^{-z}$, and $r < \Re(z) \iff s < e^{-r}$. This is a power series with radius of convergence $1$ (using e.g. Ratio test), and $e^{-r} < 1$ if $r > 0$, so it does converge uniformly.
For #2 you can use $$ \left|\frac{2^k}{z^k + z^{-k}}\right| \le \frac{2^k}{|z|^{-k} - |z|^k} \le 2^{k+1} |z|^k \ \text{if}\ |z| < 1/2\ \text{and}\ k \ge 1$$