0.1 Radius of Convergence
The radius of convergence of a power series is the radius of the largest disk at the center of the series in which the series converges. It is either a non-negative real number or \(\infty\).
When it is positive, the series [[absolute-convergence|converges absolutely]] an uniformly on compact sets inside the open disk of radius equal to the radius of convergence, and it is the Taylor series of the analytic function to which is converges.
Consider a power series \(f\) defined as \[ f(z)=\sum_{n=0}^\infty c_{n}(z-a)^n, \] where: \(a\) is a complex constant (the center of the disk of convergence); \(c_n\) is the \(n\)-th complex coefficient; and \(z\) is a complex variable.
The radius of convergence \(r\) is a non-negative real number of \(\infty\) such that the series converges if \(|z-a|<r\) and diverges if \(|z-a|>r\).