## Section8.6Convergence of Power Series

Most often, we use a power series for real values of its independent variable, but actually 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 8.5, the resulting series is valid for $\vert z\vert\le 1\text{,}$ i.e. inside a circle with radius $1\text{.}$