RadiusOfConvergenceOfAComplexFunction 2004-10-03 01:01:23 2004-10-05 02:04:30 Theorem % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here 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.''