 In our previous video, we introduced the Ultrain Series Test and actually provided a proof for the said Alternate Series Test. What I wanna do now is actually look at some examples of how we can use the Ultrain Series Test to determine the convergence of some series. Now remember, to use the Ultrain Series Test, we have some Ultrain Series, so it switches between positive, negative, positive, negative. And what we need is, if you ignore the negative signs and look at the absolute sequence, we need that that sequence is decreasing towards zero. And if that happens, our Ultrain Series would be convergent. And so let's look at some examples. Perhaps the poster child to consider here is the so-called Alternate Harmonic Series, the Alternate Harmonic Series. Because if we ignore the negative sign, this is just the harmonic series, N equals one to infinity of one over N here. But with the negative factor, so this thing will make it become positive, negative, positive, negative, positive, negative, notice here, if we expand this thing, this will look like one minus one-half plus one-third, minus one-fourth, plus one-fifth, minus one-sixth, et cetera. This number right here, we're actually gonna see is gonna get a finite number despite the fact that the harmonic series is divergent. Because if we take the absolute value here, we ignore the Alternate Factor, right? The fact that absolute value is basically gonna get rid of this guy right here. And we're left with just the sequence one over N. Now, does this sequence decrease? Is it a decreasing sequence? And the answer, of course, is yes. Because if you consider the sequence one over N plus one, if you compare that to one over N, making the denominator get bigger makes the fraction get smaller. So one over N plus one is smaller than one over N. So yes, this is a decreasing sequence. Is it, does it converge to zero? Converged to zero here. And the answer is, again, yes, the sequence one over N will approach zero. And so then by the Alternating Series Test, we see that this is a convergent series. It's convergent by this Alternating Series Test. I'll often call this AST for short. It's a convergent series. So we have just two simple tests that we have to do to show that the Alternating Harmonic Series is convergent, although the Harmonic Series is divergent. The Alternating Factor actually helps. Because kind of what's happening here is if we were to look at this graphically, let's say like the x-axis right here just for a moment. This is our x-axis. And as you take this sum, right, you add things together, where you're gonna start off with your first term, which is positive, so you get this B1. Then you take, then you're gonna add, then you subtract B2 from it. You get something like down here. Then you're gonna add B3 to it, so you get good up here. But the thing is you don't add back more than when you took away since the sequence is decreasing. Then you're gonna take something away and then you're gonna add something back. Then you're gonna take something away. Then you're gonna add something back. Then you're gonna take something away and then it kind of plays this game of jump, jump above and below, and this will decrease over time. And so what happens is you're gonna see that there's some number line kind of sitting in the middle here, that this sum is just gonna jump above and below, above and below, above and below. And that number is the limit here, this S value. And so it's really kind of nice this Alternating Harmonic Series can be convergent by the Alternating Series test. It's a lot easier for an Alternating Series to be convergent than as opposed to just a positive series because the negatives can cancel out some of the positive values here. So now let's consider the sequence here where B sub n, this is our absolute sequence, the absolute value of the a n's here. If we ignore the negative signs, we're gonna get three n over four n minus one. Does this sequence satisfy the conditions of the Alternating Series test? So does it decrease towards zero? Well, the first thing to notice here is that this sequence right here is a balanced rational function. It looks like three n over four n. So as you take the limit as n goes to infinity, this thing is gonna look like three fourths. That's not zero. So it turns out that you can't use the Alternating Series test to determine convergence. Now this is something that I want to make very clear about the Alternating Series test. The Alternating Series test can never show divergence. The Alternating Series test never shows divergence. I said that twice now. The Alternating Series test can only show convergence if the assumptions are satisfied. Now one thing important about the Alternating Series test is that if you look at the second condition, if condition two fails, that this sequence does not go towards zero, then in this situation that if you fail the second condition, then what I would actually recommend to you is then use the divergence test. Because the divergence test says that if a series is convergent, then it must have been that the sequence went to zero. So if that sequence doesn't go to zero, your series is divergent. And so that's actually a real, the divergence test and the Alternating Series test are good friends because if the second condition fails, then that actually activates the divergence test. So the fact that this sequence doesn't go to zero, actually tells us that, oh, this series is divergent. This is a divergent series. Now it's not divergent by the Alternating Series test. It's divergent by the divergence test. So do know that distinction. The two things kind of go hand in hand. And typically speaking, if an Alternating Series is divergent, it's because it's divergent by the test for divergence. Now there is a potential exception going on here. So as an example, notice that you could have a situation like the following. Here's again our x-axis. And suppose our Alternating Series that, sorry, let's look at the Alternating Sequence. Let's say a sequence does the following thing. You have the first term and the next term gets smaller. So it looks like it's decreasing, but then it decides to increase again and then it gets smaller and then smaller. And then it decides to increase again and get smaller and smaller. And then it gets increased again and smaller and smaller. And then increase again, smaller and smaller. So what you're gonna see here if you draw the dots is that it decreases and then it gets this increase and it decreases and then it gets an increase and then it decreases and then you get an increase and then decrease and it kind of does something like that. Or another way of drawing is what if your sequence kind of behaves in the following way? So it's oscillating, but what if the amplitude of the oscillation dampers over time, right? So this would be something like the function y equals e to the negative, negative x sign. Sign of x, that's not exactly true because I think does go below the x-axis. You could potentially add some quantity to it, but just sort of as a suggestion, what if we are oscillating above the x-axis but the oscillation dampers with time? So this would be an example of a sequence that does converge towards zero, right? It does converge towards zero, but maybe it's not decreasing because with these convergence tests, like the comparison test, the integrals test and the alternating series test, in order to apply them, it only has to be eventually true. Like if our sequence was like, hey, I'm doing whatever I want, but then eventually it's like, okay, I will decrease towards zero, right? Something like that. In that situation, it's eventually a positive decreasing sequence going towards zero. That would be okay. That absolute sequence we could apply the alternating series test towards. But this one, because of this oscillation happens forever, you couldn't use that. So the alternating series test would not apply in this situation. But the good news is I don't believe once we'll ever see this example because this one's a little bit harder to determine. Not all of the tests that we will learn about convergence in this class, it's not exhaustive. There are other series that can be difficult to describe. We're gonna do the more simpler ones. So the more simple ones grammar there, whoops. Let's do one more example as we conclude this video here. So let's take the series where it equals one to infinity. It's alternating because you see this negative one to the N plus one there. And then you have this series right here, N squared over NQ plus one. So we just have to figure out, does this thing decrease towards zero? I actually recommend looking at the limit as N goes to infinity first here, N squared over NQ plus one. If you look at just the leading terms on top and bottom, you're gonna end up with N squared over N cubed, which this thing will look like the limit as N goes to infinity of one over N, that does converge towards zero. So the first check is done. And in fact, this right here is telling me that my sequence N squared over N cubed plus one, this is approximately the same thing as the harmonic sequence one over N. And so this sequence does, it does decrease towards zero. And so because these things are asymptotically the same, I know for a fact that the sequence in question here will be eventually, it will eventually, it will eventually decrease towards zero. And so because of that observation, the alternating series test implies that this series is convergent. So we don't necessarily have to get into all the grime and muck about the decreasing in zero, because it just has to eventually be decreasing towards zero here. Now one could go through the details of calculating the derivative right, because if F of X equals X squared over X cubed plus one, then it's derivative by the quotient rule. I'm gonna kind of jump over the steps there. You can get that this will be two X minus X to the fourth over X cubed plus one to the fourth there, a squared, excuse me. This thing would be your first derivative as the bottom will always be positive. Where does the top go to zero? When is it less than zero? Two X minus X to the fourth is less than zero. And so as you try to like factor that thing, you get X times two minus X cubed is less than zero. So you get X, it's gonna switch sides at zero. That's a number to consider. You get like the cube root of two. And so you can see that this sequence, with a little bit more analysis that I'm omitting here, is that when X is greater than the cube root of two, you're gonna see that this sequence is, that the derivative is negative. It's also true when you're less than zero, but that's not an option. Given that we have to start at one right here. But when you get past the cube root of two, with the cube root of two is one point something. By the time you hit two, it is our sequence, our derivative will be negative, and hence the sequence is decreasing. And so that's what I was saying earlier about eventually, right? So I really, like we introduced with the limit comparison test, this idea of asymptotic comparison, right? A limit comparison. N squared over N cubed plus one is essentially the same thing as one over N when things get big. And so in terms of eventually, N squared over N cubed plus one will be eventually the same thing as one over N with a very small margin of error. So it'll eventually be decreasing and it'll eventually go to zero. And so use those asymptotic limit calculations we did with the limit comparison test to also help you out here. Because by means to be proper, we could do this derivative calculation, but this is a mess. When we can actually see asymptotically that these two are gonna be the same and this one was much easier to do. We did it above on this very video right here. So use the alternating series test to determine the convergence of the series compared to the harmonic sequence here to determine that it's decreasing towards zero. And you actually do need to check these assumptions are valid because if the assumptions are not valid, then we don't actually know if it's convergent or not. We have to check the assumptions to make sure the alternating series test actually applies in this context.