I understand that, for a vector space $V$ with scalar field $K$, the linear span of a family of vectors $S\subseteq V$ is usually defined as $$ \mathrm{sp}(S):=\left\{ \sum_{i=1}^k a_iv_i \mid k\in \mathbb{N}, a_i \in K, v_i\in S\right\} $$ In words: the span is the set of finite linear combinations of the vectors in $S$.
My question is about the motivation for defining the span to be a finite linear combination. Is there some uncomfortable conclusions that would result from using a definition like $$ \mathrm{sp}(S):= \left\{ \sum_{v\in S} a_v v \mid a_v \in K \right\}$$ I suspect that this has something to do with preserving important structures of $S$ in the span of $S$ (perhaps topological structures?), but I have no concrete examples.
One thing to keep in mind: the axioms of a vector space provide a binary sum operation, and so you get linear combinations of the form $$a_1 v_1 + a_2 v_2 = \sum_{i=1}^2 a_i v_i $$ By applying the associative law of addition you get a well-defined trinary or 3-fold sum operation, and so you get linear combinations of the form $$a_1 v_1 + a_2 v_2 + a_3 v_3 = \sum_{i=1}^3 a_i v_i $$ Continuing by induction on $n$ (and applying the associative law at each step of the induction) you get a well-defined $n$-fold sum operation for any natural number $n$, and so you get linear combinations of the form $$a_1 v_2 + a_2 v_2 + ... + a_n v_n = \sum_{i=1}^n a_i v_i $$ But there's nothing in the axioms of a vector space which lets you jump to infinite sums.
As the commenters have pointed out, you can get infinite sums by adding extra structure such as a norm which gives you a concept of convergence.
But norms are not unique, and unless you are lucky, different choices of a norm give you different concepts of convergence, and so you either have to choose a norm, or you're out of luck.
Granted, there are lots of situations in mathematics where a natural choice of norm presents itself. Hilbert spaces come to mind as an example of this phenomenon, and there are lots of useful infinite sums in a Hilbert space.
If you are very very lucky then the concept of convergence is independent of the norm. Actually you don't have to be all that lucky, because for any finite dimensional vector space over $\mathbb R$ and $\mathbb C$ the concept of convergence (the topology) is indeed independent of the norm; that covers lots of useful vector spaces.
But in general different norms give different concepts of convergence, and you should not expect infinite linear combinations to make sense independent of the choice of norm.