Power Series is Termwise Integrable within Radius of Convergence

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Theorem

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

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


Then:

$\displaystyle \int_a^b \map f x \rd x = \sum_{n \mathop = 0}^\infty \int_a^b a_n x^n \rd x = \sum_{n \mathop = 0}^\infty a_n \frac {x^{n + 1} } {n + 1}$


Proof

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

From Power Series Converges Uniformly within Radius of Convergence, $\map f x$ is uniformly convergent on $\set {x: \size {x - \xi} \le \rho}$.

From Real Polynomial Function is Continuous, each of $\map {f_n} x = a_n x^n$ is a continuous function of $x$.

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

$\displaystyle \int_a^b \map f x \rd x = \sum_{n \mathop = 0}^\infty \int_a^b a_n x^n \rd x$


The final result follows from Integral of Power.

$\blacksquare$


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