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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 (): Let ∑n=0∞an(x−x0)n\sum_{n=0}^\infty a_n(x-x_0)^n be a with radius of convergence R>0R>0, and define

f(x)=∑n=0∞an(x−x0)n(∣x−x0∣<R).f(x)=\sum_{n=0}^\infty a_n(x-x_0)^n \qquad (|x-x_0|<R).

Then for every xx with ∣x−x0∣<R|x-x_0|<R,

∫x0xf(t) dt  =  ∑n=0∞ann+1(x−x0)n+1.\int_{x_0}^{x} f(t)\,dt \;=\; \sum_{n=0}^\infty \frac{a_n}{n+1}(x-x_0)^{n+1}.

Moreover, the series ∑n=0∞ann+1(x−x0)n+1\sum_{n=0}^\infty \frac{a_n}{n+1}(x-x_0)^{n+1} is a power series with the same radius of convergence RR, and its derivative equals ff on ∣x−x0∣<R|x-x_0|<R.

This is justified by together with for the .