Power Series is Termwise Integrable within Radius of Convergence

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Theorem

Let $\displaystyle f \left({x}\right) := \sum_{n \mathop = 0}^\infty a_n \left({x - \xi}\right)^n$ be a power series about a point $\xi$.

Let $R$ be the radius of convergence of $S$.


Then:

$\displaystyle \int_a^b f \left({x}\right) \ \mathrm dx = \sum_{n \mathop = 0}^\infty \int_a^b a_n x^n \ \mathrm dx = \sum_{n \mathop = 0}^\infty a_n \frac {x^{n+1}} {n+1}$


Proof

Let $\rho \in \R$ such that $0 \le \rho < R$.

From Power Series Converges Uniformly within Radius of Convergence, $f \left({x}\right)$ is uniformly convergent on $\left\{{x: \left|{x - \xi}\right| \le \rho}\right\}$.

From Polynomial is Continuous, each of $f_n \left({x}\right) = a_n x^n$ is a continuous function of $x$.

Then from Definite Integral of Uniformly Convergent Series of Continuous Functions:

$\displaystyle \int_a^b f \left({x}\right) \ \mathrm dx = \sum_{n \mathop = 0}^\infty \int_a^b a_n x^n \ \mathrm dx$


The final result follows from Integral of Power.

$\blacksquare$


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