Radius and Interval of Convergence of a Power Series

What Convergence Means

A power series is an infinite series whose terms contain powers of a variable. A common form is:

The important question is not simply whether the series converges or diverges. Instead, we want to know:

• For which values of \(x\) does the series converge?
• For which values of \(x\) does it diverge?

For most power series, there is a central interval where the series converges and values outside that interval where it diverges. The two quantities you need to find are the radius of convergence and the interval of convergence.

Radius vs. Interval of Convergence

The radius of convergence, \(R\), tells you how far the convergence extends from the center of the power series.

If the series is centered at \(x=c\), then:

• If \(|x-c|
• If \(|x-c|>R\), the series diverges.
• If \(|x-c|=R\), you must test the endpoints separately.

This gives the key distinction:

The radius tells you the distance from the center. The interval tells you the actual \(x\)-values for which the series converges.

The Standard AP Calculus Method

When you are asked to find the radius and interval of convergence, use this four-step process:

1. Apply a convergence test, usually the Ratio Test.
2. Solve the resulting inequality to find the values where the series converges absolutely.
3. Identify the radius and candidate endpoints.
4. Test each endpoint separately. The endpoint tests determine whether the interval uses parentheses or brackets.

The most important habit is this: never stop after finding the radius. The endpoints are separate problems.

Using the Ratio Test

For a series \(\sum a_n\), the Ratio Test examines:

For a power series, the goal is to simplify this limit until it produces an inequality involving \(x\).

If the limit is less than 1, the series converges. If it is greater than 1, the series diverges. When the limit equals 1, the Ratio Test is inconclusive.

Worked Example 1: Finding the Radius and Interval

Worked Example 2: One Endpoint Converges

Worked Example 3: Alternating Endpoint

How to Read the Endpoint Results

Once you have the candidate interval, each endpoint gets its own convergence test.

• Both endpoints converge: use brackets on both ends.
• Both endpoints diverge: use parentheses on both ends.
• Left converges and right diverges: use a bracket on the left and a parenthesis on the right.
• Left diverges and right converges: use a parenthesis on the left and a bracket on the right.

The radius is always the distance from the center to either candidate endpoint. The interval is the more precise answer because it tells you exactly which endpoints are included.

Why the Endpoints Are Special

The Ratio Test typically gives you an inequality such as:

But when \(x\) is exactly an endpoint, the ratio-test limit becomes \(1\). That means the Ratio Test is inconclusive.

This does not mean that the series necessarily diverges. It means you need another test.

At the endpoints, you may encounter familiar series such as:

• Geometric series
• \(p\)-series
• Harmonic series
• Alternating series
• Comparison or Limit Comparison situations

Common Mistakes

A Fast AP Exam Strategy

When you see "find the radius and interval of convergence", think:

1. Find the Ratio Test limit.
2. Make the limit less than 1.
3. Solve for \(x\).
4. Read off the radius and candidate endpoints.
5. Plug in the left endpoint.
6. Plug in the right endpoint.
7. State the final interval using brackets or parentheses.

That workflow is reliable because it separates the problem into two parts: the Ratio Test finds the interior, and endpoint tests finish the job.

Practice Problems