I am working through exercises in Lang's Complex Analysis 3e, and have a problem..
Chapter 5, Section 3, Problem 1: Show that the following series define a meromorphic function on $\mathbb{C}$ and determine the set of poles, and their orders.
(a) $\displaystyle\sum_{n=1}^{\infty}\frac{(-1)^n}{n!(n+z)}~~~~~~$(b) $\displaystyle\sum_{n=1}^{\infty}\frac{\sin(nz)}{n!(n^2+z^2)}~~~~~~$(c) $\dfrac{1}{z}+\displaystyle\sum_{\substack{n\neq0\\n=-\infty}}^{\infty}\Bigg[\frac{1}{z-n}+\dfrac{1}{n}\Bigg]$
Now each term $f_n$ of (a) is holomorphic everywhere except at $z=-n$ which is a simple pole. These will carry over to be simple poles in the sum, at all of the negative integers. Similarly, (b) has simple poles at each non-zero integer, (c) has simple poles at all of the integers.
How do we conclude that an infinite sum of meromorphic functions is meromorphic? Do I need to show that they converge? Why is this enough?
Also, if the numerator of (b) were $\sin(\pi z)$ would this change the order/existence of the poles?