What is the Radius of Convergence of a Power Series?

The radius of convergence of a power series defines the size of the interval or disc within which the power series converges. More precisely, it's a non-negative real number or ∞ that represents how far from the center of the series the variable can be for the series to still converge.

Here's a more detailed explanation:

Understanding the Radius of Convergence

Consider a power series of the form:

∑_n=0^∞ c_n(z - a)ⁿ

where:

• z is a complex variable
• a is the center of the power series
• c<sub>n</sub> are the coefficients of the series

The radius of convergence, denoted by R, is a non-negative real number (or infinity) such that the power series converges if |z - a| < R and diverges if |z - a| > R.

Key Properties and Interpretations:

• Convergence Inside the Radius: For any complex number z whose distance from the center a is less than R (i.e., |z - a| < R), the power series converges. This region is often referred to as the disc of convergence.
• Divergence Outside the Radius: For any complex number z whose distance from the center a is greater than R (i.e., |z - a| > R), the power series diverges.
• On the Boundary ( |z - a| = R ): The behavior of the power series on the boundary of the disc of convergence (i.e., when |z - a| = R) is more complex. The series may converge at some points on the boundary, diverge at other points, or converge conditionally. Further analysis is required to determine convergence on the boundary.
• R = 0: If R = 0, the power series converges only at the center z = a.
• R = ∞: If R = ∞, the power series converges for all complex numbers z.

Methods for Determining the Radius of Convergence:

1. Ratio Test: The ratio test is a common method:

   R = lim_n→∞ |c_n / c_n+1|

   If this limit exists.

2. Root Test: The root test is another useful method:

   R = 1 / lim_n→∞ |c_n|^1/n

   (assuming the limit exists). If the limit is 0, then R = ∞. If the limit is ∞, then R = 0.

3. Holomorphic Function Approach: 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. This method is especially useful when the power series represents a function known to have singularities.

Examples:

• Example 1: Consider the power series ∑_n=0^∞ zⁿ. Applying the ratio test: R = lim_n→∞ |1/1| = 1. The series converges for |z| < 1 and diverges for |z| > 1. On the boundary |z| = 1, the series diverges.

• Example 2: Consider the power series ∑_n=0^∞ zⁿ/n!. Applying the ratio test: R = lim_n→∞ |(1/n!) / (1/(n+1)!)| = lim_n→∞ (n+1) = ∞. The series converges for all z. This power series represents the exponential function e^z, which is holomorphic everywhere.

• Example 3: The Taylor series for 1/(1-z) centered at z=0 is ∑_n=0^∞ zⁿ. This function has a singularity at z=1. Therefore, the radius of convergence is |1-0| = 1.

In Summary:

The radius of convergence is a critical concept in the study of power series, indicating the range of values for which the series converges. Understanding how to determine and interpret the radius of convergence is essential for working with power series representations of functions.