Understand that some power series only converge for certain values of x. The range of these values of x is called the interval of convergence. The radius of convergence is half of the length of the interval of convergence.
Determine the interval of convergence of a power series by applying the ratio test and testing the endpoints of the interval.
Quick Lesson Plan
In this lesson, students explore three series written in terms of a variable, x, and determine their convergence. What students will discover is that the conditions for convergence depend on the value of x. This can be seen in a simple way for geometric series where the common ratio must be less than 1, and then extended to other series using the ratio test. This lesson will hopefully be an “aha!” moment for your students as they discover that their prior knowledge of geometric series and the ratio test can be used to determine convergence of many other series that are not geometric.
In question 1, students will encounter a geometric series that contains terms with the variable x. Although their previous study of geometric series involved constants only, this example will build a connection between those series and the infinite polynomials they encountered in 10.11. Encourage your students to see that there is a common ratio between terms that can be written in terms of x. Questions 1c and 1d will allow them to explore the idea that for some x values, convergence occurs and for others, divergence occurs. Use the monitoring questions below to guide them to eventually have a general rule for which x values lead to convergence.
The next part of this lesson will prompt them to think about non-geometric series and to consider how to test their convergence using the ratio test. The series in question 2 will result in convergence for all x values, but the series in question 3 will require students to remember that convergence occurs when the limit of the ratio is less than 1. The goal is for students to see that performing this ratio test is identical to the one for a series of constant terms, but that sometimes the limit found will include x. The question will then arise: how do we know when convergence occurs when using the ratio test?
What patterns do you notice in the first series? Have you ever seen any series like this before?
Do you think the value of x influences convergence/divergence? How?
How can you know if a geometric series converges?
What convergence test might work for power series (especially containing factorials or exponential terms)?
How can you create an interval from the limit you find?
Why is it important to have absolute value around the ratio?
Once students begin to see that the ratio test allows them to obtain an interval of all x values that result in convergence, you can discuss what this means in a broader sense: it is possible that power series converge for certain x values, but not others. The final step of this process is to discuss the endpoints of the interval obtained. Do they result in convergence? By substituting each endpoint into the original series, students will discover a series whose convergence they can analyze using their previous study of series. If the series converges for that value of x, that endpoint should be included. Students must repeat this process for both endpoints in order to write an interval of convergence with the proper inequalities. The radius of convergence seems logical, then, to be half the length of the interval found.
Historically, when a question involving the interval or radius of convergence appears on a free response question, it generally earns 4-5 points. For this reason, this is a topic that students should consider a “must learn”! Although the process of finding the interval is lengthy, remind students that often points can be earned just for knowing what to do. Did they set up a ratio correctly? Did they attempt to find a limit (even if the limit is wrong)? Did they try to test endpoints (even if they didn’t determine the result correctly)? Sometimes, points can be earned by communicating the process, not only by finding the correct result. By scaffolding the practice you provide your students, they will gradually feel comfortable with the process and will find the desired interval consistently.