Қуат қатарларының жуықталу радиусы

Domain of convergence of power series
In mathematics, 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 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. In case of multiple singularities of a function (singularities are those values of the argument for which the function is not defined), the radius of convergence is the shortest or minimum of all the respective distances (which are all non negative numbers) calculated from the center of the disk of convergence to the respective singularities of the function.

Usually, in scientific applications, only a finite number of coefficients are known. Typically, as increases, these coefficients settle into a regular behavior determined by the nearest radius limiting singularity. In this case, two main techniques have been developed, based on the fact that the coefficients of a Taylor series are roughly exponential with ratio where r is the radius of convergence. The basic case is when the coefficients ultimately share a common sign or alternate in sign. As pointed out earlier in the article, in many cases the limit exists, and in this case Negative means the convergence limiting singularity is on the negative axis. Estimate this limit, by plotting the versus , and graphically extrapolate to (effectively ) via a linear fit. The intercept with estimates the reciprocal of the radius of convergence, This plot is called a Domb–Sykes plot. The more complicated case is when the signs of the coefficients have a more complex pattern. Mercer and Roberts proposed the following procedure. Define the associated sequence Plot the finitely many known versus , and graphically extrapolate to via a linear fit. The intercept with estimates the reciprocal of the radius of convergence, This procedure also estimates two other characteristics of the convergence limiting singularity. Suppose the nearest singularity is of degree and has angle to the real axis. Then the slope of the linear fit given above is Further, plot versus , then a linear fit extrapolated to has intercept at .

A power series with a positive radius of convergence can be made into a holomorphic function by taking its argument to be a complex variable. The radius of convergence can be characterized by the following theorem:

The radius of convergence of a power series f centered on a point a is equal to the distance from a to the nearest point where f cannot be defined in a way that makes it holomorphic. The set of all points whose distance to a is strictly less than the radius of convergence is called the disk of convergence. The nearest point means the nearest point in the complex plane, not necessarily on the real line, even if the center and all coefficients are real. For example, the function

has no singularities on the real line, since has no real roots. Its Taylor series about 0 is given by

The root test shows that its radius of convergence is 1. In accordance with this, the function f(z) has singularities at ±i, which are at a distance 1 from 0. For a proof of this theorem, see analyticity of holomorphic functions.