M-test continuity and integration corollary

Let $X$ be a set and let $f_n:X\to\mathbb{R}$ (or $\mathbb{C}$). Suppose:

• each $f_n$ is continuous on $X$ (when $X$ is a metric space ), and
• there exist $M_n\ge 0$ with $|f_n(x)|\le M_n$ for all $x\in X$, and $\sum M_n$ converges .

Corollary:

• The series $\sum f_n$ converges uniformly on $X$ (Weierstrass M-test ).
• Hence its sum $f=\sum f_n$ is continuous (uniform limit of continuous functions ).
• If $X=[a,b]$ and each $f_n$ is Riemann integrable , then $\int_a^b \sum_{n=1}^\infty f_n(x)\,dx=\sum_{n=1}^\infty \int_a^b f_n(x)\,dx$ (term-by-term integration).

Connection to parent theorems: Combine the Weierstrass M-test with the uniform limit theorem for continuity and the uniform convergence-and-integration theorem .