Interchanging limit and integral

Under uniform convergence, the limit of Riemann integrals equals the Riemann integral of the limit.

Interchanging limit and integral

Interchanging limit and integral: Let $f_n:[a,b]\to\mathbb{R}$ be Riemann integrable for every $n$ , and suppose $f_n\to f$ uniformly on $[a,b]$ . Then $f$ is Riemann integrable and

\lim_{n\to\infty}\int_a^b f_n(x)\,dx \;=\; \int_a^b f(x)\,dx.

This is a basic continuity property of the Riemann integral with respect to uniform convergence, and it is the key step behind term-by-term integration of uniformly convergent series .