### 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.

#### Teacher Journal: CA I.3

• Geometric Series  Students generally find this activity a little easier than what we have just been working on. The main point where they struggle is the algebraic manipulation required for Q2 and the limit work that is needed to find the formula that geometric series converge to. We also find that some students will do Q5 without using the formula discovered in Q4 - instead they will use Mathematica to find the sum. It is important to come back to Q5 after they have worked on it and point out that the formula will give them the same answer and that they should be able to use it as they may not have Mathematica available.
Posted Jun 15, 2012, 8:53 AM by Mairead Greene
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