Understand that comparing the ratio of consecutive terms in a sequence can provide insight into whether the series converges or diverges.
Simplify an expression for the ratio of the n+1 term and the nth term of a sequence and evaluate the limit as n approaches infinity.
Apply the ratio test to determine convergence or divergence.
Quick Lesson Plan
In today’s activity, students will analyze series by determining the ratio of consecutive terms and use this to determine convergence or divergence. In questions 1 and 2, students look at geometric sequences and series which they should be very familiar with from Algebra 2 and Precalculus. Students should recall that geometric series with a common ratio less than 1 will have a finite sum! If students are struggling to remember this, use an intuitive approach to help students reason through the value of the infinite sum. For example, by finding just a few partial sums in question 1, students will quickly realize that the sum is getting bigger and bigger and has no hope of converging to a finite value. In question 2, students should see that the term being added each time is getting smaller and smaller, making the contribution to the sum almost negligible for terms with large values of n. If your school uses the Math Medic Precalculus lessons, this would be a great time to bring up Bernie the Chicken!
In question 3 students encounter a new sequence that is not geometric but nevertheless simple to work with. After writing the first few terms of the sequence, students calculate the ratio from a term to its previous term. They note that this ratio is not constant (since the sequence is not geometric), and should also notice that the ratio is decreasing. Writing the nth term and the n+1 term is an important skill for this lesson, so make sure students feel comfortable constructing these algebraic expressions and knowing what they represent.
In question 5, students must simplify an expression in terms of n. Students may need a few concrete examples of simplifying factorial expressions to be able to generalize patterns for how these expressions simplify. Having students write out 4! as 4*3*2*1 and 5! as 5*4*3*2*1 will help them see which factors remain after dividing terms.
Question 6 gets at the idea of a limit and question 7 has students use their analysis to make a conjecture about the convergence of the series. Note: students do not yet formally know the conclusions of the ratio test so expect their reasoning to be more informal and intuitive. It is likely that students will use their understanding of geometric series to conclude that the series diverges if the ratio is less than 1. If groups do not arrive at this conclusion, do not worry. The whole class debrief will make explicit how the ratio test can be applied.
What do you notice about how this sequence is growing? Is there a common difference? A common ratio?
What does it mean that this series diverges? How do you know it will diverge?
Can you find patterns in the ratios even if the ratio is not constant?
What does n! mean?
How does the ratio of terms depend on the term number? What happens to the value of the ratio as the term numbers get bigger and bigger?
When debriefing the activity, use ratio notation in the margins of questions 1 and 2 to build the connection between geometric series and other types of series that are analyzed by their ratios of consecutive terms. After going over the final aspect of the ratio test in question 7, go back and do the ratio test for the geometric sequence in question 2 shown in the right hand margin of the answer key.
The ratio test is one of the most useful convergence tests and is the foundation for future topics such as the interval and radius of convergence. The ratio test is particularly helpful when the general term of a series involves factorials or exponents as these simplify nicely when comparing consecutive terms.
Note that the ratio test is ONLY helpful when the value of the limit is less than 1 (the series converges) or greater than 1 (the series diverges). If the value of the limit is equal to 1, the test is inconclusive and a different test must be applied. When the value of the limit is 1, the ratio test does not prove or disprove convergence or divergence!
Note also that the sign of the ratio of consecutive terms does not matter. The test uses the absolute value of the ratio of the n+1 term to the nth term to express the magnitude of the ratio, not its sign.
The most challenging aspect for students when using this test is the algebraic manipulation of simplifying the ratio of the n+1 term to the nth term. This will require practice so students can build fluency and intuition about which terms cancel and how to appropriately rewrite an expression. Taking the limit as n approaches infinity of this simplified ratio is a great review of early Calculus topics! (See Topic 1.14–Limits at Infinity).
The ratio test shows up on the Calc BC exam every year. Make sure that students clearly communicate their process and intention so they can earn partial credit even if their algebraic manipulations include errors. Beware of linkage errors by being explicit about when an expression represents the ratio of terms versus the limit as n approaches infinity of the ratio of the terms.
Because this type of question is often worth 3-4 points, it is important that students are aware of what work earns those points. For example, setting up the ratio correctly is often the first point and using correct limit notation earns the second point. Reviewing FRQs from previous exams with students will help them to see exactly what work is expected and how it has been scored in the past. Most importantly, remind them that communication is one of the Mathematical Practices and for that reason, showing each step clearly is of paramount importance.