I'm having some troubles understanding when a series of $L^1_\text{loc}$ functions $\sum_n f_n$ converges uniformly on compact sets. Is there a theorem stating that if the $f_n$ (or their absolute values) satisfy a certain condition, then the series is uniformly convergent on compact sets? Something like Weierstrass' $M$-test for total convergence. I've been looking for it in several books and forums but I've not been able to answer my question.
Uniform convergence on compact sets: sufficient conditions
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sequences-and-series
uniform-convergence