Weierstrass M-test

Suppose functions $f_n$ on $E$ satisfy $|f_n(x)|\le M_n$ for every $x\in E$, with $M_n\ge 0$ and ${\sum }_n{M}_n<\infty$. The Weierstrass M-test says that ${\sum }_n{f}_n$ converges absolutely and uniformly on $E$. Indeed every tail is bounded uniformly by the numerical tail ${\sum }_{n>N}{M}_n$, which tends to zero. For example, ${\sum }_{n\ge 1}{x}^n/n^2$ converges uniformly on $[-1,1]$ by $M_n=1/n^2$. The bound must hold over the whole chosen domain, not depend on its point. Uniform convergence then preserves continuity when the summands are continuous.