Weierstrass_M-Test

Corollary

For each $n \in \mathbb{N}$, let $f_{n}$ be a function defined on a set $A \subseteq \mathbb{R}$, and let $M_{n} > 0$ be a real number satisfying

$$ \left| f_{n}\left. (x) \right. \right| \leq M_{n} $$

for all $x \in A$. If $\sum_{n = 1}^{\infty}M_{n}$ converges, then $\sum_{n = 1}^{\infty}f_{n}$ converges uniformly on $A$.

Proof