Do the terms of an infinite series constitute a countable set? Given an infinite series (e.g. trigonometric expansion, exponential, whatever)  $\sum_{\infty}T_{n}$, were one to consider the terms of this series as the members of a set $S$, it is obvious that the set would be an infinite one (given that the terms come from an infinite series in the first place).
My question is would this set be considered countably infinite or uncountably infinite? 
My guess is toward countably infinite, since each member of the set (i.e. term $T_{m}$ in the series) can be uniquely mapped to the corresponding real number $m \in \mathbb{Z}$ and thus there exists a bijection between the set $S$ and $\mathbb{Z}$, and hence by the definition of a countable set, the infinite series should be countably infinite. But being painfully aware of my tendency to jump to easy conclusions, I would like someone better educated to confirm this.

If I understand correctly your question, you construct from a formal series $\sum_{i \in I} T_i$ a set $S = \{T_i | i \in I\}$. 
Then the cardinal of $S$ is less than or equal to the cardinal of $I$, almost by definition. 
In particular, if $I = \mathbb{N}$, or if $I = \mathbb{Z}$, then $S$ is countable. 
PS : note that it is not necessarily infinite, for instance if $T_i = T_j$ for any $i,j \in I$, then $|S|=1$. 
