We are given positive functions $f_n(t)$ with limits (as $t \rightarrow 0$) $\overline{f}_n$.
Consider the following statements
$f_n(t) \rightarrow \overline{f}_n$ as $t \rightarrow 0$ for all $n$.
$\sum_{n \geq 1} f_n(t) < \infty$ for all $t$ and $\sum_{n \geq 1} \overline{f}_n < \infty$
Do these imply
$\sum_{n\geq 1} f_n(t) \rightarrow \sum_{n\geq 1} \overline{f}_n$ as $t \rightarrow 0$?
If they do not what additional conditions are needed.
Let $f_n = 1_{({1 \over n+1},{1 \over n}]}$, $\bar{f}_n = 0$.
Then $\sum_{n \ge1} f_n = 1_{(0,1]}$, $\sum_{n \ge 1} \bar{f}_n =0$.
If you want a strictly positive counter example, consider $f_n+{1 \over 2^n}, \bar{f}_n+{1 \over 2^n}$.