 Welcome to the GVSU Calculus Screencasts. In this episode, we're going to talk about using the divergence test. To begin, recall that the nth partial sum of the series is just the sum of the first n terms of the series. And the partial sums form a sequence, we call the sequence of partial sums. And if that sequence of partial sums converges to sum number s, then we say that the series converges. Whereas if the sequence of partial sums diverges, then we say that the series itself diverges. In this screencast, we're going to look at a test that will allow us to determine when certain types of series diverge. Consider the example where we have the series k over k plus 1, as k goes from 1 to infinity. The nth partial sum, s sub n, of this series is the sum of the first n terms. The k equals 1 term is a half, the k equals 2 term is 2 thirds, k equals 3 term is 3 fourths, and so on. And so s sub n is 1 half plus 2 thirds plus 3 fourths, all the way up to n over n plus 1. Our goal is to determine if this sequence of partial sums converges or diverges. So pause the video for a moment and calculate the first 5 of these partial sums. And based on that information, I can conjecture as to whether you think this sequence of partial sums converges or diverges. Then resume the video when you're ready. The partial sums s sub 1 through s sub 5 are calculated as follows. s1 is just the k equals 1 term, 1 half. s2 is the sum as k goes from 1 to 2 of k over k plus 1. When k is 1, we get a half. When k is 2, we get 2 thirds. Notice this is just s1 plus 2 thirds. So we get 7 sixth, which is about 1.67. s3 is the sum as k goes from 1 to 3. So that'd be the k equal 1 term plus the k equal 2 term plus the k equal 3 term. And notice that that's just the same as s2 plus the k equal 3 term, 3 quarters. So we end up with 23 twelfths, which is about 1.92. Similarly, s4 is just s3 plus the k equal 4 term, which gives us to about 2.72. And s5 is s4 plus the k equal 5 term, 5 sixth, which gives us to about 3.55. These numbers seem to be increasing, but it's not quite clear what happens to the sequence of partial sums. It's hard to generalize from just 5 terms. So let's calculate some of these partial sums for larger values of n. The 100th through 104th partial sums are shown here. We're getting numbers that are 95, 96, 98, 99. They seem to be increasing as n gets big. Notice that when n gets close to 100, the difference between the nth partial sum and the n minus first partial sum is about 1. And this happens because the nth partial sum is the sum of the terms a sub 1, a sub 2 up through a sub n minus 1 plus a sub n. And if we subtract from that the n minus first partial sum, a sub 1 plus a sub 2 up through a sub n minus 1, notice that all the first n minus 1 terms are the same, so they'll just cancel out, leaving us with a sub n. So in our example, s sub 101 minus s sub 100 is just a sub 101. That's the term when k equals 101, that's 101 divided by 102, which is really close to 1. Now if we were to look at the sequence of partial sums graphically, this picture seems to indicate that the sequence of partial sums is just constantly increasing as n goes to infinity. Moreover, it looks like this plot is linear as n gets large. And this happens because, as you'll recall, s sub n minus s sub n minus 1 is just a sub n, which in this case is n over n plus 1, and n over n plus 1 is close to 1 if n is big, and the bigger n gets, the closer n over n plus 1 gets to 1. So the sequence, the graph of the sequence becomes more and more linear as n gets really big. And that means that the sequence of partial sums diverges as n goes to infinity. The conclusion we ultimately draw from all of this is that the series k over k plus 1 as k goes from 1 to infinity diverges. Now the general idea that was indicated by our example is this. Suppose we have a series, sum a sub k, as k goes from 1 to infinity. If s sub n is the nth partial sum of the series, then the difference between the n minus first partial sum and the nth partial sum is just a sub n. So if the series a sub k converges, then the sequence of partial sums has to converge. And this means that the limit of s sub n minus s sub n minus 1 as n goes to infinity is the limit of a sub n as n goes to infinity. And since s sub n and s sub n minus 1 are both partial sums of the same series, they have to converge to the same thing. So the limit of the difference is zero. So we conclude that if the series a sub k converges, then it must be the case that the sequence of nth terms, a sub n, converges to zero. Now the way we use the divergence test is just the opposite of what we argued a minute ago. If the sequence a sub n of nth terms doesn't converge to zero, then the series itself of a sub k has to diverge. Because we just argued that if the series a sub k diverges, then the sequence a sub n of nth terms has to converge to zero. And this gives us the divergence test. If the sequence a sub n of nth terms doesn't converge to zero, then the series a sub k has to diverge. Now there's something really important to note about this test. This test can only tell us that a series diverges. It can never tell us that a series converges. That's why we call this the divergence test. If the sequence of nth terms a sub n actually converges to zero, then this test doesn't apply. The divergence test tells us nothing about the behavior of the series a sub k. The only way the divergence test applies is if the sequence of nth terms doesn't converge to zero. Okay, so let's test our knowledge of the divergence test. Pause the video for a moment and look at these two series. What does the divergence test tell us about them? Resume the video when you're ready. In the first case, note that the sequence of nth terms, n squared plus three over two n cubed plus two, converges to zero. That means the divergence test doesn't apply. The divergence test doesn't tell us anything about the corresponding series. In the second case, the limit of the sequence of nth terms to n to the fourth plus five n over n to the fourth plus seven is two. And that's not zero. And if the limit of the sequence of nth terms isn't zero, then the divergence test applies and tells us that the corresponding series diverges. So to summarize, the divergence test is the test that can be used to tell us if a series diverges. And if the limit of the sequence of nth terms isn't zero, then the series, the corresponding series, has to diverge. However, if the limit of the sequence of nth terms is zero, then the divergence test just doesn't apply. It doesn't tell us anything about the corresponding series, a sub k. Well, that concludes our screencast on the divergence test. We hope you'll come back and visit again soon.