Radius of convergence
In mathematics, the radius of convergence of a power series is the radius of the largest disk in which the series converges. It is either a non-negative real number or ∞displaystyle infty . When it is positive, the power series converges absolutely and 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 it converges. Contents 1 Definition 2 Finding the radius of convergence 2.1 Theoretical radius 2.2 Practical estimation of radius in the case of real coefficients 3 Radius of convergence in complex analysis 3.1 A simple example 3.2 A more complicated example 4 Convergence on the boundary 5 Rate of convergence 6 Abscissa of convergence of a Dirichlet series 7 Notes 8 References 9 External links Definition For a power series ƒ defined as: f(z)=∑n=0∞cn(z−a)n,displaystyle 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