The Direct Comparison Test - Calculus 2

In the various tests for convergence of a series we have looked at so far (geometric series, p-series, and the integral test), the terms of the series needed to be pretty simple and have special characteristics in order for each test to be applied. Any slight deviation from these characteristics will result in not being able to use those tests for a series.

So when we encounter a series that looks similar to a geometric series, p-series, or series we can use the integral test for, what we can do in order to determine if they converge or diverge is use one of two other tests known as comparison tests. The first comparison test is called the Direct Comparison Test, and is the focus of this lesson. The second comparison test is called the Limit Comparison Test, and we will save that one for the next lesson.

What these comparison tests will allow us to do is identify features in a particular series that are similar to a special type of series whose converge we do know, and compare those series to determine the convergence of the original series. The Direct Comparison Test deals with "directly" comparing two series based on the whose terms are bigger or smaller, and the Limit Comparison Test deals with using a limit to compare two series. In each case, one series will be the series with more complicated terms whose convergence is unknown, and the other series will be the comparison series that we choose, whose convergence we do know. 

Using these comparison tests may sound complicated at first, but it is not as difficult as it initially seems. As long as you are able to identify features of a p-series and geometric series, using comparison tests are actually fairly simple. We will first focus on the Direct Comparison Test, which leads us to the key points of this lesson.

In this lesson, you will learn:

• What the Direct Comparison test is and why it works
• How to decide when to compare a series using the direct comparison test 
• How to choose a comparison series and use it to show that another series converges or diverges

I hope you find this video to be helpful!

-Josh