This section introduces a special type of series that will be particularly important to us as we continue in Chapter 8. It’s unusually easy to determine whether an Alternating Series converges or diverges.

Basic Learning Objectives

These are the tasks you should be able to perform with reasonable fluency when you arrive at your next class meeting. Important new vocabulary words are indicated in italics.

  • State the definition of an alternating series, especially the meaning of the sequence of positive terms.

  • Be able to identify an alternating series and its sequence of positive terms, as well as tell when a series is not alternating.

  • Be able to state the Alternating Series Test and apply it in simple cases.

Advanced Learning Objectives

In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:

  • Be able to apply the Alternating Series Test to come to a correct conclusion.

  • Be able to define conditionally convergent and absolutely convergent and use the terms properly in a sentence.

  • Use appropriate convergence tests to determine the behavior of conditionally or absolutely convergent series.

  • Choose the most appropriate convergence test for a given series.

To prepare for class

Use these resources to become proficient with the basic objectives (see above) before class:

  • Read Section 8.4 in Active Calculus. You may choose to only skim the Alternating Series Estimation Theorem for now, and then review it in more detail after class.

  • Watch the following videos (by GVSU Math):

  • Do the Preview Activity for this section (on WeBWorK if required by your teacher).

  • Watch the following videos (by GVSU Math):

After Class

  • Watch the following video (by Dr. Trefor Bazett) about the Riemann Rearrangement Theorem, and the difference between absolutely and conditionally converging series:

  • Download (and print/treasure) this handout containing a summary of the most important facts & tests about series (excerpt from Active Calculus by Matthew Boelkins), as well as a decision flowchart for picking the best convergence test (by Ralph Freese).

  • Watch the following video (by Dr. Trefor Bazett) which shows 8 different series and explains how to choose which Convergence Test to apply to each:


Charles Fortin Avatar Charles Fortin
Gabriel Indurskis Avatar Gabriel Indurskis


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