= Uniform absolute convergence
{title2=$\sum_n|f_n|\text{ converges uniformly}$}
A <series> of <functions> is uniformly absolutely convergent on a <set> if the series of their <absolute values> has <uniform convergence> there. A sufficient condition is the <Weierstrass M-test>: if $|f_n(x)|\leq M_n$ for every $x$ and $\sum_n M_n<\infty$, then the tail is bounded uniformly by $\sum_{n\geq N}M_n\to0$. This also proves <uniform convergence> of the original <series> by the <Uniform Cauchy criterion>.
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