Understand that a sum of infinite terms can have a finite value.
Define convergent and divergent series.
Determine convergence for geometric, harmonic, and p-series.
Quick Lesson Plan
Today we introduce students to the world of series by building on what they already know: geometric series. In the first part of the activity, students consider a scenario where 3 people split a pizza that is continually cut into fourths. The original idea for this intro was given to us by Tony Record and we appreciate his willingness to let us build an EFFL lesson around it!
In questions 1-3, students reason informally about the size of each slice (written as a fraction of the total pizza) and the total amount of pizza eaten by each person after each slice. This distinction is of course the difference between the nth term of a geometric sequence and the nth partial sum of the sequence. In question 4, students think about what happens if this process continues indefinitely, i.e. an infinite sum. What we love about this intro is that it is entirely intuitive for students that each person would eventually eat ⅓ of a pizza, since they are splitting it equally. Students are able to understand why an infinite sum can have a finite value, which is a key understanding of the lesson.
In question 5, we attach some sequence notation to the ideas students explored earlier in the activity, building the bridge to geometric series and the sum formula a/1-r. Note that students do not have to know this formula to make it through the first page of the activity. This formula will be reviewed in the margin notes when debriefing the first page of the activity.
After debriefing the first page of the activity, have students continue on to page 2, completing all three parts of question 6. Note that while the sequences 1/n and 1/n^2 are not geometric, students can use inductive reasoning to notice patterns in their sums. In question 6c, we hope some students will think back to our lesson on indefinite integrals when we explored the integral of 1/x and 1/x^2 from x=1 to x=∞. Note that the value of the definite integral is NOT the same as the sum of the series; however, the same intuition can be used to explain why 1/n^2 converges whereas 1/n diverges.
What fraction of the pizza is the second slice? The third slice? How do you know?
How can you find the total pizza eaten by one person after each slice?
Is it possible for infinite slices of pizza to give you a finite amount of food? HOW?!
What is happening to the terms of these sequences?
What is happening with the partial sums of these sequences?
What differences do you notice between how 1/n is changing and how 1/n^2 is changing?
The goal of today’s lesson is to both (re)introduce the idea of a series and to build a series toolkit of the most basic 3 series: the geometric series, the harmonic series (1/n) and the p-series (1/n^p). Students should feel very comfortable determining convergence if a series takes on one of these three forms. Be prepared that the value of p in the p-series could be written as a fraction, decimal, or irrational number. Students will see this in question 2 of the Check Your Understanding.
Note that students may initially confuse geometric series with p-series since both have exponents. The important distinction is that geometric series represent exponential functions (y=b^x) where the variable/term number is the exponent, representing repeated multiplication by a factor. P-series represent a power function (y=x^b) where the exponent is fixed and the variable/term number is the base. The value of p is fixed for each series, it is not a variable!
Later tests like the Direct Comparison Test and the Limit Comparison Test will rely on students’ familiarity with these three toolbox series.
When students encounter a series on the BC exam, a good first step is to determine if the series is one from their toolbox. This will allow them to quickly determine convergence without spending a lot of time choosing a test and doing the associated calculations. Check out the full flowchart here!
Questions involving series convergence can appear as multiple choice or free response questions. When the format is multiple choice, often students are asked to identify if a series converges or diverges. Sometimes this is presented in the I, II, III format (choices of I only, I and II, all three, etc). However, when these questions appear in an FRQ, it is important that students communicate the requirements for using the test just as they would for the theorems in calculus. Once the requirements are stated and the test is performed, an appropriate conclusion would look like “This series converges/diverges by the _____ Test”. Getting in the habit of responding this way will ensure that maximum points are earned, regardless of the specific question.