Consider a doubly-indexed sequence $(a_{n,k})$ which, for each $k$, converges as $n\to\infty$ to an absolute constant $C$. In what generality can we say that for a sequence $b_k$ $$\lim_{n\to\infty}\sum_{k=0}^n a_{n,k}b_k = C\sum_{k=0}^\infty b_k \quad?$$
Motivation: I am doing a combinatorics homework assignment and we have been asked to prove something related to $q$-Pochhammer symbols, so my question arose specifically from a very nice notion of convergence (formal power series), and an extremely simple $b_k$: $$ a_{n,k}=\binom{2n}{n+k}_q \to (q;q)_\infty \qquad\qquad b_k =x^k.$$
I'm more or less convinced that I can do this in my example, because you can reason that the $x$ and $q$ are algebraically independent and so you can just get away with comparing coefficients. This is basically a slightly more advanced version of "convergence in each coordinate".
However, actually proving this always comes down to one or two steps that I am sure are true but feel rather uncomfortable stating. I know that one can interpret a formal power series ring as a topological vector space, and I think the gold standard would be to prove it in this setting (take $a_{n,k}$ to be scalars and $b_k$ to be vectors) for any sequence $b_k$. Perhaps we would need to restrict it to a sequence with convergent series; that's fine.
To me this does seem rather likely. I know it's generically dangerous to pull out a limit from a summation sign, and topological vector spaces are particularly weird. But the fact that the constant is universal (not dependent on $k$) makes me feel that some sort of uniformity may save the day.