Series & The Divergence Test - Calculus 2

An important application of sequences is how they can be used to represent infinite summations, or what we call infinite series, commonly shortened to just "series."

To put it simply, a series is the sum of the terms of a sequence. For example, if we have a sequence represented by the nth term 3n, the terms of the sequence would be 3, 6, 9, 12, ... and so on. However, the series for 3n would be equal to the sum of those terms: 3 + 6 + 9 + 12 + · · · .

Similar to sequences, we are interested in determining the convergence of series. We want to know what value, if any, does the series converge to, or in other words, what is the sum of the series as we add more and more terms? We can determine this convergence by analyzing what we call partial sums.

As the name implies, partial sums are just part of the entire sum for an infinite series. We can create a sequence of these partial sums and use that sequence to determine the convergence of the series. (More on that in the video)

However, that method of using partial sums is not easily applicable to all series. Most series will require the use of other methods to determine their convergence (or divergence). These different methods will be the focus of the next several lessons in Calculus 2. One of those methods is known as The Divergence Test, and is the only other method we will look at in this specific lesson.

The Divergence Test is a quick way to determine if a series diverges by using a limit. The specifics of this test will be discussed in the lesson video, and so let's get right into it!

In this lesson, you will learn:

• What an infinite series is
• How to find the nth partial sum of a series and use it to determine the convergence of the series
• How to use The Divergence Test to quickly determine if a series diverges

I hope you find this video to be helpful!

-Josh