How to Calculate Radius and Interval of Convergence

Understanding the radius and interval of convergence is essential for analyzing the behavior of power series. This guide explains how to calculate these values, including the step-by-step process, practical examples, and common pitfalls.

What is Convergence?

A power series is an infinite sum of terms that can be written in the form:

Σ (from n=0 to ∞) aₙ (x - c)ⁿ

The convergence of a power series refers to the values of x for which the series sums to a finite value. The radius of convergence (R) is the distance from the center c within which the series converges. The interval of convergence is the set of all x values for which the series converges, which may include the endpoints.

Calculating the Radius of Convergence

The radius of convergence can be found using the ratio test, which compares the absolute values of consecutive terms. The formula is:

R = lim (n→∞) |aₙ / aₙ₊₁|

If the limit exists and is finite, then R is the radius of convergence. If the limit is zero, the radius is infinite. If the limit does not exist or is infinite, the radius is zero.

For many common power series, the ratio test yields a simple formula. For example, the geometric series Σ (xⁿ) has a radius of convergence R = 1.

Determining the Interval of Convergence

Once the radius is known, the interval of convergence is determined by checking the endpoints of the interval (c - R, c + R). The series may converge at one or both endpoints, or it may diverge at both.

To check the endpoints, substitute x = c + R and x = c - R into the series and determine convergence using other tests such as the nth term test or direct substitution.

Endpoint Test	Result
If lim (n→∞) \|aₙ Rⁿ\| = 0, the series converges at x = c + R	Converges
If lim (n→∞) \|aₙ Rⁿ\| = ∞, the series diverges at x = c + R	Diverges

Practical Examples

Example 1: Geometric Series

Consider the series Σ (xⁿ). Using the ratio test:

R = lim (n→∞) |xⁿ / xⁿ⁺¹| = lim (n→∞) |1/x| = 1

The interval of convergence is (-1, 1). Checking the endpoints:

At x = 1: Σ (1ⁿ) diverges (harmonic series)
At x = -1: Σ ((-1)ⁿ) diverges (alternating harmonic series)

Example 2: Exponential Series

For the series Σ (xⁿ / n!), the ratio test gives:

R = lim (n→∞) |(xⁿ / n!) / (xⁿ⁺¹ / (n+1)!)| = lim (n→∞) |(n+1)/x| = ∞

The series converges for all x, so the interval of convergence is (-∞, ∞).

Frequently Asked Questions

What does a radius of convergence of zero mean?

A radius of zero means the series only converges at its center point. This occurs when the ratio test yields an infinite limit.

Can the interval of convergence include infinity?

Yes, if the radius of convergence is infinite, the interval of convergence is (-∞, ∞), meaning the series converges for all real numbers.

How do I know if a series converges at an endpoint?

Use the nth term test or direct substitution to check the behavior of the series at the endpoints. If the limit of the terms is zero, the series converges; otherwise, it diverges.