Uniform convergence and integration

Uniform convergence and integration: Let $f_n:[a,b]\to\mathbb{R}$ be Riemann integrable for all $n$ , and let $s_N=\sum_{n=1}^N f_n$ be the partial sums. If $s_N\to s$ uniformly on $[a,b]$ , then $s$ is Riemann integrable and

\int_a^b s(x)\,dx \;=\; \lim_{N\to\infty}\int_a^b s_N(x)\,dx \;=\; \sum_{n=1}^\infty \int_a^b f_n(x)\,dx,

where the series of real numbers on the right converges.

Remarks

In applications, uniform convergence of $(s_N)$ is often established using the Weierstrass M-test.