Why the Endpoints Matter
When you find the interval of convergence of a power series, the ratio test usually gives you an open interval where the series converges. But the ratio test typically tells you nothing about the two endpoints.
Those endpoints must be tested individually. One endpoint might converge while the other diverges, both might converge, or both might diverge.
For example, suppose your work gives the interval
You are not finished. You still need to determine what happens at and .
The Basic Strategy
Once the radius of convergence has given you candidate endpoints, use this four-step process:
- Find the radius of convergence, usually using the ratio test.
- Identify the two endpoints of the resulting open interval.
- Substitute each endpoint into the original series. Do not simply plug the endpoint into the ratio-test limit.
- Use an appropriate convergence test on each resulting series.
The key idea is that each endpoint is a separate series problem. The convergence behavior at one endpoint has no reason to match the behavior at the other.
Why You Cannot Stop at the Radius
Consider the power series
The ratio test gives a radius of convergence of , so we initially know that the series converges for
But this is only an open interval. The points and are still undecided.
At this stage, writing would be incomplete. You have to test both endpoints.
Worked Example 1: Both Endpoints Converge
Find the interval of convergence of
Worked Example 2: One Endpoint Converges and One Diverges
Find the interval of convergence of
Worked Example 3: Both Endpoints Diverge
Consider
Which Test Should You Use at an Endpoint?
The ratio test is usually what gets you to the endpoints, but it is often not the best tool for testing the endpoints themselves.
After substitution, look at the resulting series and identify its familiar pattern.
- Geometric series: converges when the common ratio has absolute value less than 1.
- p-series: converges when .
- Alternating series: consider the Alternating Series Test.
- Harmonic series: diverges.
- Terms do not approach zero: use the nth-Term Test for divergence immediately.
- More complicated series: consider comparison, limit comparison, ratio, or another appropriate convergence test.
The Endpoint Decision Table
Once both endpoints have been tested, the final interval follows directly:
Think of the two endpoints as two independent yes-or-no decisions. The left endpoint can be included or excluded regardless of what happens at the right endpoint.
A Fast AP Exam Workflow
On an AP Calculus BC problem, you can make endpoint testing much more efficient by separating the problem into two phases.
Phase 1: Find the candidate interval
- Apply the ratio test.
- Solve the inequality that makes the ratio-test limit less than 1.
- Write the resulting open interval.
Phase 2: Test the endpoints
- Plug the left endpoint into the original series.
- Identify the resulting series.
- Determine whether it converges or diverges.
- Plug the right endpoint into the original series.
- Use a convergence test again.
- Write the final interval using parentheses or brackets appropriately.
Common Mistakes
The radius only tells you how far the interval extends from the center. It does not automatically tell you whether either endpoint is included.
Never assume that both endpoints behave the same way. One can converge while the other diverges. Always substitute and test them separately.
The ratio test often produces a limit of 1 at the endpoints. That means the ratio test is inconclusive there. Go back to the original series and use another appropriate test.
When testing an endpoint, substitute the endpoint into the original power series. Do not substitute it into an inequality or into a simplified expression from the ratio test and assume that determines convergence.
If an endpoint converges, use a bracket [ or ]. If it diverges, use a parenthesis ( or ). The notation should match your endpoint tests.
Practice Problems
Find the interval of convergence of .
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Find the interval of convergence of .
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Find the interval of convergence of .
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A power series has radius of convergence and center . At the left endpoint, substitution produces the harmonic series. At the right endpoint, substitution produces an alternating harmonic series. What is the interval of convergence?
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