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Course: Calculus III
Topic: Sequences and Infinite Series
Subtopic: Alternating Series

Overview

Alternating series are ones where the signs on the terms in the series alternate signs (eg. positive, negative, positive, etc.) They can be quite interesting and often have surprising results. For instance, divergent series where the terms are all positive are not necessarily divergent when the signs alternate. The alternating series test is an excellent choice for determining the convergence or divergence of an alternating series.

Objectives

By the end of this topic you should know and be prepared to be tested on:

• 11.8.1 Recognize if a series is alternating
• 11.8.2 Write an alternating series in summation notation
• 11.8.3 Know that the Alternating Harmonic Series is (conditionally) convergent to ln2
• 11.8.4 Understand the conditions necessary to apply the alternating series test and know the types of series for which the alternating series test is a good choice
• 11.8.5 Know how to use the alternating series test accurately to determine the convergence or divergence of an alternating series
• 11.8.6 Understand when the alternating series test is inconclusive
• 11.8.7 Understand the remainder in the alternating series test and be able to use the remainder formula to find it for a given alternating series

Terminology

Define: alternating series, alternating series test, remainder in alternating series

Supplemental Resources (optional)

Paul's OL Notes - Calc II: Alternating Series Test

Selwyn Hollis's Video Calculus:
Alternating Series and Absolute Convergence

Patrick JMT Just Math Tutorials:
Alternating Series
Alternating Series - More Examples
Alternating Series - Estimation Theorem

More tutorial videos if you need them are listed at James Sousa's MathIsPower4U - Calc II. Scroll down middle column to "Infinite Series" for related titles.