Let $D\subset \Bbb{C}$ be non-empty, open, and $\{f_n\}_{n=1}^{\infty}$ a sequence of meromorphic functions on $D$. In Henri Cartan's complex analysis book, the following definition is given:
We say the series $\sum f_n$ converges uniformly on a subset $A\subset D$ if there is an $n_0\in \Bbb{N}$ such that for all $n\geq n_0$, $f_n$ has no poles in $A$, and the series $\sum_{n\geq n_0}f_n$ converges uniformly on $A$.
While I understand the definition, I was wondering if there is a way to phrase the definition "more naturally". What I mean is that usually, one starts by defining concepts for sequences of functions $\{f_n\}$, and then defines similar concepts for series by applying the sequential definition to the partial sums $s_n:= \sum_{i=1}^nf_i$. So, there's a nice "symmetry" in the definitions for sequences and series, but by the way the definition is phrased above, it seems this "symmetry" is broken because one first requires the existence of an $n_0\in\Bbb{N}$ such that blablabla.
So, I'm asking if it is possible to first define the concept of "uniform convergence of a sequence of meromorphic functions on a subset $A\subset D$", and then somehow naturally defining the concept of "uniform convergence of a series of meromorphic functions on a subset $A\subset D$".