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Section 7.9 Convergence of Power Series

Most often, we use a power series for real values of its independent variable, but they work perfectly well for complex variables. It is actually easier to study the set of values for which a power series is valid, called its region or circle of convergence, if we think of it as a function of a complex variable. If a function has a power series expansion around some point \(a\text{,}\) then the circle of convergence extends to the nearest point at which the function is not analytic. (Briefly, a function which is not analytic is singular in some way. A function is certainly not analytic at any point at which its value becomes infinite or at a branch point of a root.) For example, the function \(\frac{1}{z^2+1}\) seems perfectly well-behaved if \(z\) is real, but blows up if \(z=\pm i\text{.}\) If we expand this function in a power series around \(z=0\text{,}\) using the binomial expansion in Section 7.7, the resulting series is valid for \(\vert z\vert\le 1\text{,}\) i.e. inside a circle with radius \(1\text{.}\)