The Limit Comparison Test - Calculus 2

In the last lesson we looked at the first comparison test for series known as the Direct Comparison Test. Upon introducing that test, I mentioned that there was a second comparison test that we will look at in the next lesson. That test is called the Limit Comparison Test, and is the focus of this lesson!

Similar to the direct comparison test, the limit comparison test will allow us to identify features in a series that are similar to a special type of series whose convergence we do know, and compare those series to determine the convergence of the original. When using the direct comparison test, this involved "directly" comparing two series based on the whose terms are bigger or smaller.

However, the limit comparison test is slightly different. Instead of comparing the series on the size of the terms, we will use a limit to compare them. Specifically, we will evaluate a limit as n approaches infinity of the sequence from the original series divided by the sequence from the comparison series. If we can show that this limit exists, then we can conclude that the series shares the same convergence or divergence as the comparison series.

Overall, there is technically not a lot of "new" math to learn here. The limit comparison test combines two skills that we are already familiar with into one convergence test. We will identify what type of series to compare to a certain series of interest, just like we did with the direct comparison test, and then we will evaluate a limit at infinity, which we have been doing since Calculus 1. Doesn't sound too bad, right? Well let's get right to it then!

In this lesson, you will learn:

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

I hope you find this video to be helpful!

-Josh