CA I.3 Geometric Series
Purpose: To define geometric series, write them in sigma notation, establish when they are finite or infinite, and if finite discover what they sum to.
Classroom Procedure: Students will work in small groups for all of this activity. We start by asking students to work on Q1 in small groups and to make sense of the definition given. This is again a mini reading activity and it is better if students try this themselves first before the instructor translates. We then give students some time to try Q2 but this may require a good deal of class discussion to complete. Then students can work on Q3 and after a quick check that everyone has written down geometric series they should continue on to Q4 in their small groups. At various times I talk to the whole class during this question to ensure that everyone understands what they are doing - I may even fill in a line of one of the tables to ensure that everyone is understanding the distinction between the three columns. Finally a class discussion is needed to establish that Q4 a and b have been completed. Although students should have read and thought about Q4c, most students will not reach the conclusions (particularly the second one) without help in a class discussion. Q5 should be done as homework.
Ideas this Activity Builds On: This builds on our work in developing sigma notation and series in the first two activities. In addition in the synthesis questions the students have just completed they worked with two geometric series although we didn't call them that at the time.
Introduction/Motivation of the Activity: To better understand series we can study particular types of series where it is easier for us to determine whether they will be finite or infinite. The first type of series that we will study in this way are geometric series.
Need to Establish by the End of Activity/Wrap-Up:
What is a geometric series
When do they converge/diverge
If they converge - what do they converge to
Special type of series because they are the only type of series where we have this much information about what they will converge to.