Group Post due on the OpenLab Sunday, November 1

Motivation

This week’s group assignment is based on HOT Topics Standards 7 & 8:

  • Standard 7: Determine whether an infinite series converges absolutely, converges conditionally, or diverges
  • Standard 8: Determine whether an infinite series converges or diverges

It will be very straightforward for you to figure out which of your HOT Topics Portfolio questions align with Standards 7 & 8: just read the instructions for the questions carefully. It may not be straightforward to figure out how to answer them, however. You will have to use a convergence test to answer each question but depending on which series you have been assigned and which tests you’ve learned so far, it might be a test that you’re already familiar with or it might be a new one that you haven’t seen yet.

One of the major goals of our unit on series is to determine which convergence test to use when. Often, especially when you are just learning to use these convergence tests, you’ll end up trying to use the wrong test before you figure out which test is the right one. You’ll probably try a few different wrong ones! A test is the right one to determine the convergence of a particular series if, when you apply it correctly, it tells you whether the series converges or diverges.

Sometimes there is more than one right test to use for a particular series, but there is always more than one wrong one! Gaining experience applying the wrong one a few times and seeing what goes wrong is an important part of the learning process. It helps you understand why the right test is the right one and helps you recognize which is the right test next time.

This week’s group assignment has two parts.

Part 1

On your own before you meet with your group:

  • determine which of your two HOT Topics Portfolio questions align with Standards 7 & 8,
  • write down a list of convergence tests (not just their names, but how each test works),
  • for each of your two HOT Topics Portfolio questions, write down three convergence tests that you might use to answer the questions (it’s okay if these are somewhat wild guesses, but do try to make informed guesses).

With your group:

  • Share your two questions with your group as well as the three convergence tests you wrote down for each of them.
  • Take a poll: do your group members think that any of your guesses are definitely wrong or definitely right?
  • As a group, discuss with each other why you made your guesses and why you think someone else’s guess is right or wrong.
  • Do you want to adjust your guesses after the discussion?

Part 2

As a group, choose one of your group members’ HOT Topics Portfolio questions. Then, as a group, choose three convergence tests: two that are definitely wrong and one that is right for this question. Work together to write down what it looks like to apply these tests to the series you chose.

Choose three group members; two who will write up what it looks like to apply the wrong test and explain why the test did not work out, and one who will write up the complete and correct solution. You will be sharing this work with the class, so—especially for the wrong tests—in words, write a description of how you figured out that this was the wrong test for the series you chose.

Choose a secretary who will submit your group’s work on the OpenLab. Title the post “Group n: Two wrong convergence tests, one right one” and select the category Week 9 group post: two wrong, one right before publishing the post. Include all group members’ names in the post.

The post should include photos of the written work for the three applications of tests; indicate clearly which tests you are applying for each as well as which two are wrong and which one is right.

You will earn participation credit for this group post.

Small detail

Just like when you were integrating, you sometimes had to use more than one method at a time (for example, many of your trigonometric substitution questions ended up requiring a further $u$-substitution later). The same is true here: you sometimes need to apply more than one convergence test, especially if you must determine whether a series that converges actually converges absolutely or conditionally. If this is the case for the question your group chose, your “one right test” will actually be two right tests, used together. Be sure to indicate this in your post.

Example of a wrong test