The Integral Test - Calculus 2

When we first introduced series, we looked a quick test that can be performed to determine if it is a diverging series. This test was the Divergence Test, and just involved taking the limit of the nth term of the series. If that limit was not equal to 0, then the series diverged! So this test was extremely helpful for determining if a series is a diverging series. However, it was not very helpful at determining if a series converges instead.

Remember that there are two different possible results from performing the divergence test. The series will either diverge, or we have absolutely no idea if it converges or diverges. The latter happens when the limit of the nth term of the series is equal to 0. When the divergence test produces this result, we are going to need to use alternative methods to determine if the series diverges or converges. The first of these methods that we are going to look at is known as The Integral Test.

In general, the integral test is most helpful when the divergence test is inconclusive and the nth term of the series is easily integrated. But before we can even think about setting up an integral, there are a few requirements that need to be met: The corresponding function (the nth term of the series but n is replaced with x) to the series needs to be positive, continuous, and decreasing for values from 1 to infinity. There is some room for adjustment when these requirements are not quite met, but generally these are the necessary features of the corresponding function in order to use the integral test.

Once we know the integral test can be used for a series, we can then integrate the corresponding function from 1 to infinity and the result of that integral will tell us if our series converges or diverges. This will be the main focus of the lesson.

In full, we will go over how to check the requirements for the integral test to be applied to a series and then how to proceed with setting up an improper integral to perform the test, as well as interpret the results from the integral and what they say about the original series. These problems can sometimes become slightly lengthy, and so let's get right into it!

In this lesson, you will learn:

• How to use the Integral Test to determine if a series converges or diverges
• How to determine if the corresponding function to a series is positive, continuous, and decreasing
• How to adjust when the requirements of positive, continuous, and decreasing are not quite met 

I hope you find this video to be helpful!

-Josh