Term-by-term integration of a power series

Inside its radius of convergence, a power series can be integrated by integrating each term.

Term-by-term integration of a power series

Term-by-term integration (power series): Let $\sum_{n=0}^\infty a_n(x-x_0)^n$ be a power series with radius of convergence $R>0$ , and define

f(x)=\sum_{n=0}^\infty a_n(x-x_0)^n \qquad (|x-x_0|<R).

Then for every $x$ with $|x-x_0|<R$ ,

\int_{x_0}^{x} f(t)\,dt \;=\; \sum_{n=0}^\infty \frac{a_n}{n+1}(x-x_0)^{n+1}.

Moreover, the series $\sum_{n=0}^\infty \frac{a_n}{n+1}(x-x_0)^{n+1}$ is a power series with the same radius of convergence $R$ , and its derivative equals $f$ on $|x-x_0|<R$ .

This is justified by uniform convergence of power series on compact subsets together with interchanging uniform limits and integration for the Riemann integral .