Radius of convergence

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In mathematics, the radius of convergence of a power series is a quantity, either a non-negative real number or ∞, that represents a domain (within the radius) in which the series will converge. Within the radius of convergence, a power series converges absolutely and uniformly on compacta as well. If the series converges, it is the Taylor series of the analytic function to which it converges inside its radius of convergence.

Contents

Definition

For a power series ƒ defined as:

where

The radius of convergence r is a nonnegative real number or ∞ such that the series converges if

and diverges if

In other words, the series converges if z is close enough to the center and diverges if it is too far away. The radius of convergence specifies how close is close enough. On the boundary, that is, where |z − a| = r, the behavior of the power series may be complicated, and the series may converge for some values of z and diverge for others. The radius of convergence is infinite if the series converges for all complex numbers z.

Finding the radius of convergence

Two cases arise. The first case is theoretical: when you know all the coefficients cn then you take certain limits and find the precise radius of convergence. The second case is practical: when you construct a power series solution of a difficult problems you typically will only know a finite number of terms in a power series, anywhere from a couple of terms to a hundred terms. In this second case, extrapolating a plot estimates the radius of convergence.

Theoretical radius

The radius of convergence can be found by applying the root test to the terms of the series. The root test uses the number

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