Radius of convergence

For a power series f defined as: where The radius of convergence r is a nonnegative real number or

[1] Two cases arise: The radius of convergence can be found by applying the root test to the terms of the series.

Note that r = 1/0 is interpreted as an infinite radius, meaning that f is an entire function.

The ratio test says the series converges if That is equivalent to Usually, in scientific applications, only a finite number of coefficients

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

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.

Its Taylor series about 0 is given by The root test shows that its radius of convergence is 1.

The arctangent function of trigonometry can be expanded in a power series: It is easy to apply the root test in this case to find that the radius of convergence is 1.

It may be cumbersome to try to apply the ratio test to find the radius of convergence of this series.

But the theorem of complex analysis stated above quickly solves the problem.

We solve by recalling that if z = x + iy and eiy = cos(y) + i sin(y) then and then take x and y to be real.

Example 1: The power series for the function f(z) = 1/(1 − z), expanded around z = 0, which is simply has radius of convergence 1 and diverges at every point on the boundary.

If h is the function represented by this series on the unit disk, then the derivative of h(z) is equal to g(z)/z with g of Example 2.

[5] If we expand the function around the point x = 0, we find out that the radius of convergence of this series is

Both the number of terms and the value at which the series is to be evaluated affect the accuracy of the answer.