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Course: Calculus III
Topic: Sequences and Infinite Series
Subtopic: Comparison Tests

Overview

There are two different comparison tests used to determine if a series is convergent or divergent. In each your prior knowledge of the convergence or divergence of a basic series is used to compare to in determining the convergence or divergence of a new series. The direct comparison test (a.k.a. simply the "comparison test") involves knowing how the new and old series compare in terms of one being "contained" in the other. The limit comparison test uses the limit of a ratio of the new and old limits.

Objectives

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

• 11.7.1 Know the convergence and divergence of some basic limits, eg. the harmonic series, so that you can use those facts in one of the comparison tests
• 11.7.2 Understand the conditions necessary to apply the direction comparison test and know the types of series for which the direct comparison test is a good choice
• 11.7.3 Know how to use the direct comparison test accurately to determine the convergence or divergence of a series
• 11.7.4 Understand when the direct comparison test is inconclusive
• 11.7.5 Understand the conditions necessary to apply the limit comparison test and know the types of series for which the limit comparison test is a good choice
• 11.7.6 Know how to use the limit comparison test accurately to determine the convergence or divergence of a series
• 11.7.7 Understand when the limit comparison test is inconclusive

Terminology

Define: comparison test, direct comparison test, limit comparison test

Supplemental Resources (optional)

Paul's OL Notes - Calc II: Comparison Test & Limit Comparison Test

Patrick JMT Just Math Tutorials:
Limit Comparison Test and Direct Comparison Test 1
Limit Comparison Test and Direct Comparison Test 2

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.