When do series converge to meromorphic functions on $\mathbb{C}~$? 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?
 A: 
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?

The answer is essentially contained in Montel's theorem. Let's say we have a sum of meromorphic functions:
$$
\sum_{n = 0}^{\infty} f_n
$$
First question: Can we find a domain $D \subseteq \mathbb{C}$ so that none of the poles of any $f_n$ is contained in $D$? If yes, then the next question is if the series
$$
F_N := \sum_{n = 0}^{N} f_n
$$
is locally bounded on $D$ (this is the part where you need to prove the convergence of the series in the sense that you need to show that there is a finite upper bound for all $z \in D$). If it is, we can apply Montel's theorem and conclude that $F_N$ is normal, that is it has a subsequence that converges compactly to a holomorphic function $F$ on $D$. Since the sequence $(F_N)$ is a Cauchy sequence, we see that actually $(F_N)$ itself converges to $F$ on $D$.
In order to show that $F$ is meromorphic on $\mathbb{C}$, we need to show that $F$ does not have an essential singularity and that the set of poles is a discrete subset of $\mathbb{C}$. 
This will work out if you can show that


*

*$F$ is finite in every z that is not a pole of any $f_n$,

*the set of all poles of all $f_n$ is discrete and

*for every z that is a pole of some $f_n$, z is a pole of only finite many $f_n$.
