 Welcome back. In our previous video, we saw the idea of absolute convergence. What does it mean for a series to be absolutely convergent? And we also introduced the idea of conditionally convergent. And we showed an example of an absolutely convergent series, which is convergent and a conditionally convergent series, which is to say that it was convergent, but not absolutely convergent. Can we come up with an example of an absolutely convergent series, which is not convergent? And it turns out the answers know that the two objects are related to each other. So in fact, if a series, take the sum of the ANs, if the series is absolutely convergent, that actually implies that the series is convergent itself. So what I mean by this is if the sum of the absolute values converges, then the sum of the original sequence must have converged as well. The series are convergent. And so let's see a little bit of why that is. So this proof here is gonna be based upon the following inequalities right here. If you take a number and you add to it its absolute value, you get one of two things, like take for example, take two plus the absolute value of negative two. One of two things are gonna happen here. Either when you take the absolute value here, you just forget the negative sign, so you get something like two plus two, which is gonna add up to be four. Notice this is two times the original number two. The other option is if your number started off as negative, when you add to it its absolute value, I guess I shouldn't be taking the absolute value negative two there. You get two plus its absolute values two, but if you take negative two plus its absolute value, you're gonna get negative two plus two, which is zero. So the basic idea here is if you add a number to its absolute value, you either double its value because it was positive or you'll get zero because it was negative. And so that gives us some inequalities to play around with. This is gonna be helpful for a forthcoming comparison test. So A plus the absolute value is greater than zero, but less than two times the absolute value. And this doesn't matter whether the sequence A is positive, negative, or zero. The terms of the sequence don't matter. So let's assume that the series is absolutely convergent. What that means is the series of absolute values is convergent. That's what absolute convergent means. So this is a convergent series, voila. Okay, we got that. Well, if the series involved the absolute values is convergent, then two times that series will also be convergent. Or if you wanna bring the two inside, the sum of two times the absolute value of A n is a convergent series. And so this series, we're gonna make a comparison onto this series right here. So by the comparison test, using the comparison test, because this series is convergent, this series has to be convergent because it's smaller than two times the absolute value of A n. So the sum of A n plus the absolute value of A n, that likewise is a convergent series. Now, wait a second, the series of A n, this is the actual more interesting, right? We assume this series was absolutely convergent. We wanna show that the series is convergent. So if the series of just the A n's, notice that it's actually a difference of two convergent series. We can take the series where you add together A n and the absolute value of A n, and we can subtract from it the series of the absolute value of A n, which was the absolute series, which we already know is convergent. When you take the differences of these two, if you take the difference of these two series, you'll end up with the sum of A n. And the difference of two convergent series is necessarily convergent. And so therefore we get that A n is convergent. So absolute convergence implies convergence. Absolute is a stronger condition than just convergence, right? We've now sort of broken up convergence into two different families. So if you have your series, right, there's two options. Your series could be convergent, but it could likewise be divergent. We've seen examples of these things, but now the convergent family can break off in the two subcategories. Convergent could end up being absolutely convergent, like we've seen before, or it could be conditionally convergent. These are the possibilities we have at hand. And so believe it or not, there are scenarios where it's easier to show that something is absolutely convergent than it is to show that it's convergent. But since absolute convergence implies convergence, we can use the absolute series to help us show convergence. Let me give you such an example. Determine the series, determine whether the series, the sum n equals one to infinity of cosine n over n squared is convergent or not. Let's determine whether this series converges. Now the issue is that this is not a positive series. It's not positive because, well, cosine of n is sometimes less than zero, right? We know for a fact that cosine can vary between one and negative one, and there's gonna be some whole numbers n for which it's gonna be less than zero. So we have some issue going on there. Sometimes cosine of n is zero for some n. So it's not a positive series, but on the other hand, it's not an alternating series. It's not an alternating series it's alternating series go positive, negative, positive, negative, positive, negative, positive, negative, et cetera, it switches every other time. One time is positive, one time is negative. One time is positive, one time is negative. But we don't see that happening with this series. The thing is cosine of n is gonna be kind of like positive, positive, negative, positive, positive, a negative, negative, negative, positive. I'm not saying those are exact numbers, but it feels much more sporadic. It's not just every other time alternating. It's a little bit less predictable when it's gonna be positive, when it's gonna be negative. So the alternating series test does not apply in that situation. So what do we do instead here? Well, the idea is to show the convergence of this series I recommend using the absolute series. So we're actually gonna, we can think of this previous theorem if we were to switch it around. So notice we see here that absolute convergence implies convergence. And so we could call this like, kind of like how we have the divergence test. We could talk about the test for absolute convergence. If we prove the series is absolutely convergent, then it's convergent. And so if a sub n is our sequence cosine of n over n squared, notice the absolute sequence, the absolute value of a n. This will be the absolute value of cosine of n over n squared. Why is that helpful? Well, n squared, since you're going from one to infinity here, n squared will always be positive. So it's just the numerator that's giving us a hang up here. But as it's cosine, right? Cosine is always bounded above by one. And so our absolute sequence is always bounded above by one over n squared. And so this can be helpful here because if you look at the absolute series, n equals one to infinity of the absolute value of cosine n over n squared, because we're taking absolute value of this thing right here, we're always gonna get positive. This thing is greater than or equal to zero. But because of the inequality we mentioned right here, this thing is less than or equal to the sum where n equals one to infinity of one over n squared, for which this is a p-series and by the p-test, because our p-value is two, this is convergent. That is, this series is convergent, the p-series. But since we have a bigger series, which is convergent, this implies that the smaller series is likewise convergent and it's convergent by the comparison test. Because we found a bigger convergent series. And then finally, since the absolute series is convergent, this implies that our original series is absolutely convergent. It's absolutely convergent by this test of absolute convergence. But heck, if it's absolutely convergent, that means it's actually convergent as well. So in this situation, believe it or not, it's actually easier to show that the series is absolutely convergent than it is to show that it's convergent. But since absolute convergence implies convergence, we get the answer that we were looking for.