Term-by-term operations on series of functions

Let $\sum_{n=1}^\infty f_n$ be a series of functions on an interval $[a,b]$.

Proposition (term-by-term integration): Suppose each $f_n$ is Riemann integrable on $[a,b]$ and $\sum f_n$ converges uniformly on $[a,b]$ to a function $f$. Then $f$ is Riemann integrable and $\int_a^b f(x)\,dx=\sum_{n=1}^\infty \int_a^b f_n(x)\,dx.$

Proposition (term-by-term differentiation): Suppose each $f_n$ is differentiable on $[a,b]$, and:

• the series of derivatives $\sum f_n'$ converges uniformly on $[a,b]$ to a function $g$, and
• the original series $\sum f_n(x_0)$ converges at some point $x_0\in[a,b]$. Then $\sum f_n$ converges uniformly on $[a,b]$ to a differentiable function $f$, and $f'(x)=\sum_{n=1}^\infty f_n'(x)\quad\text{for all }x\in[a,b].$

These statements formalize the usual calculus manipulations with function series; the uniform convergence hypotheses are the key analytic input.

Proof sketch: Integration: apply the “uniform limit of integrable functions ” theorem to partial sums $s_N=\sum_{n=1}^N f_n$. Differentiation: apply the “uniform convergence and differentiation ” theorem to the sequence of partial sums $s_N$, noting $s_N'= \sum_{n=1}^N f_n'$ and using the convergence at $x_0$ to pin down constants of integration.