Radius of Convergence

The concept of radius of convergence is used in the field of complex analysis to determine the set of values for which a power series converges. In the context of the given question, we are interested in finding the set of values of z for which the function X(z) attains a finite value.

To understand the concept of radius of convergence, let's consider a power series representation of a function X(z):

X(z) = ∑(n=0 to ∞) a_n * (z - z_0)^n

Here, X(z) is a function of the complex variable z, and a_n is the nth coefficient of the series. The term (z - z_0)^n represents the powers of z centered around the point z_0.

The radius of convergence, denoted by R, is the distance from the center point z_0 to the nearest point where the series converges. In other words, it defines the boundary beyond which the series diverges.

Convergence and Divergence

When the magnitude of z - z_0 is less than the radius of convergence (|z - z_0| < r),="" the="" series="" converges="" and="" x(z)="" attains="" a="" finite="" value.="" on="" the="" other="" hand,="" if="" |z="" -="" z_0|="" /> R, the series diverges and X(z) goes to infinity or does not have a finite value.

Calculating the Radius of Convergence

To find the radius of convergence, we can use the ratio test or the root test. These tests involve taking the limit of the ratio or the root of the absolute values of consecutive terms in the series.

Let's consider the ratio test for simplicity. According to the ratio test, if the following limit exists:

lim(n→∞) |a_(n+1)/a_n| = L

Then the radius of convergence can be calculated as:

R = 1/L

If L = 0, it implies that the series converges for all values of z, and the radius of convergence is infinite.

If L = ∞, it implies that the series only converges at the center point z_0, and the radius of convergence is zero.

If L is a finite positive value, then the series converges for |z - z_0| < r,="" where="" r="" />

Therefore, the set of all values of z for which X(z) attains a finite value is determined by the radius of convergence of the power series representation of X(z).