Practice Problem Set: Power Series and Intervals of Convergence

How to Approach These Problems

A power series is an infinite series whose terms contain powers of a variable. A typical power series centered at has the form

For a given power series, there is usually an interval of values of for which the series converges and values outside that interval for which it diverges. The endpoints require special attention because the convergence behavior can change at exactly those two values.

The AP Calculus BC Strategy

1. Use the Ratio Test to find the radius of convergence. 2. Convert the radius into an open interval. 3. Test the two endpoints separately. 4. State the final answer using interval notation.

Worked Examples

Example 1: Find the Radius and Interval of Convergence

Find the interval of convergence of .

Step 1: Apply the Ratio Test to the absolute values of consecutive terms. Let . Then .

Step 2: Simplify the limit.

Step 3: The Ratio Test gives convergence when . Therefore, .

Step 4: Test the endpoints separately: .

Endpoint 1: At , the series becomes , which is a geometric series with ratio . It converges.

Endpoint 2: At , the series becomes . This is the harmonic series, so it diverges.

Result:

Example 2: Both Endpoints Require Testing

Find the interval of convergence of .

Step 1: Use the Ratio Test: .

Step 2: Convergence requires , giving the open interval .

Step 3: Test . The series becomes , which is a p-series with . Therefore, it converges.

Step 4: Test . The series becomes , which is the alternating harmonic series. By the Alternating Series Test, it converges.

Result:

Example 3: A Series Centered Away from Zero

Find the interval of convergence of .

Step 1: Apply the Ratio Test: .

Step 2: The condition for convergence is , so .

Step 3: The candidate endpoints are .

Step 4: At , the series becomes , which converges by the p-series test.

Step 5: At , the series becomes , which diverges because it is a constant multiple of the harmonic series.

Result:

Example 4: Recognizing Endpoint Behavior

Find the interval of convergence of .

Step 1: The Ratio Test gives .

Step 2: Therefore the radius of convergence is , giving candidate endpoints .

Step 3: At , the series becomes . This converges by the Alternating Series Test.

Step 4: At , the series becomes . Since the terms do not approach zero, the series diverges.

Result:

The Most Important Endpoint Rules

The Ratio Test usually gives you the radius of convergence, but it generally does not determine what happens at the endpoints. Once you have the radius, plug each endpoint into the original series and use an appropriate convergence test.

Geometric series: converges when .
p-series: converges when .
Alternating Series Test: check that the terms decrease toward zero.
nth-Term Test: if the terms do not approach zero, the series diverges immediately.
Comparison Test: useful when an endpoint produces a familiar benchmark series.

Mistake: Stopping after the Ratio Test

If the Ratio Test gives , you do not yet have the interval of convergence. You only know that the series converges inside the interval and diverges outside it. The two endpoints must still be checked individually.

Mistake: Using the Same Endpoint Test Automatically

The left and right endpoints can behave completely differently. One endpoint may converge while the other diverges, so never assume that both endpoints should be included—or excluded—without testing them.

Mistake: Forgetting the Center of the Series

A series centered at with radius has endpoints . The radius tells you the distance from the center; it is not automatically the right endpoint of the interval.

Practice Problems

Question 1

Find the radius and interval of convergence of .

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Question 2

Find the interval of convergence of .

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Question 3

Find the radius and interval of convergence of .

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Question 4

Find the interval of convergence of .

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Question 5

Find the interval of convergence of .

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Question 6

Find the radius and interval of convergence of .

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Question 7

Find the interval of convergence of .

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Question 8

Find the radius and interval of convergence of .

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AP Exam Checklist

Before You Move On

Can you identify the center of a power series?
Can you use the Ratio Test to find the radius of convergence?
Can you convert the radius into candidate endpoints?
Can you substitute each endpoint into the original series?
Can you recognize geometric, p-series, and alternating-series behavior?
Can you report the final answer correctly in interval notation?

On an AP Calculus BC free-response question, the most important habit is to separate the problem into two stages: first find the radius, then investigate the endpoints. A correct radius with incomplete endpoint analysis is not a complete interval-of-convergence answer.

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