I'm hoping to find a lot of information for the case when we can say:
$$\lim_{a \to 1}{ \sum_{k=b}^c{f(a,k)} } = \sum_{k=b}^c{ \lim_{a \to 1}f(a,k) }$$
...or, more generally, when $a$ approaches some value from both sides.
So, essentially, when can we exchange limits and summations?
If you have only finitely many summands, then the above is true if you read it this way: Assuming that all limits on the RHS exist, then the limit on the LHS exists, too, and this equiality holds. You can derive it by induction using the result for sum of two functions: Proving the limits of the sum of two functions is equal to the sum of the limits
If you are interesting also in a similar question for (countably) infinitely many summands, you can find some posts on this site which discuss this, for example, Limit Summation interchanging or Under what condition we can interchange order of a limit and a summation?