30B10-RadiusOfConvergenceOfAComplexFunction.tex

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Let $f$ be an analytic function defined in a disk of radius $R$ about a point $z_0 \in \mathbb{C}$.  Then the radius of convergence of the Taylor series of $f$ about $z_0$ is at least $R$.

For example, the function $a(z) = 1 / (1 - z)^2$ is analytic inside the disk $|z| < 1$.  Hence its the radius of covergence of its Taylor series about $0$ is at least $1$.  By direct examination of the Taylor series we can see that its radius of convergence is, in fact, equal to $1$.

Colloquially, this theorem is stated in the sometimes imprecise but memorable form ``The radius of convergence of the Taylor series is the distance to the nearest singularity.''
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