Integrating Term by Term

Theorem

Consider the power series
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Let $C$ be a simple piecewise smooth curve which lies inside the circle of convergence. Then we can integrate the power series term by term:
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Proof

The function $f(z)$ defined by the power series is continuous on $C$, so the integrals in () are well-defined. We need to show that


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Since $C$ lies inside the circle of convergence, the series converges uniformly on $C$ to $f(z)$. For any $\epsilon $, there is an $N(\epsilon )$ so that, for all $z$ on $C$,
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By the triangle inequality for integrals and the above inequalities, for $n\geq N,$
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Since $\epsilon $ is arbitrary, the limit in () is zero.

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