 Welcome back to our lecture series, Math 1220 Calculus II for students at Southern Utah University. As usual, I'll be your professor today, Dr. Andrew Misseldine. This lecture, which is lecture 32, we very appropriately refer to as Street Fighter II. And that's, the reason for that is for those who've been following this series, when we had learned about techniques of integration, there was a lecture, I think it was like lecture 17 or something like that in our series. I can't remember off the top of my head. Maybe I'll put a comment down below if I've ever figured it out. But we had had a lecture which referred to as Street Fighting Integrals. And the idea was, when someone's not telling you, use this integration by parts, use trigonometric substitution. You're left on your own to fight for yourself. That is you're fighting in the street fight for your life. You can use whatever tools you have with you. If you know how to use integration by parts and it works, use it. If you know how to use Taekwondo, then use it, right? The same is also applicable for series. So, if you're reading this thing below here, right? We can see that the convergence of series can be somewhat difficult to do at times. And therefore, it can be challenging to identify the right series test to use. And so in this, which will correspond to lecture, well, section 11.7 in James Stewart's textbook, I want to first mention some strategies one could use for testing convergence of series. And then after this, we're gonna see a lot of teeny tiny short videos where we do one series step by step where I want the viewer actually to pause the video and try it out for themselves before they continue and see the solution I'm gonna offer. But what are the tools that we can use as we try to do these convergence tests? Now, I'm listening in pretty much the order that I think about them, right? First on the docket here would be the test for divergence. That a series could be divergent because as you take the series of adding up the ANs, if the sequence of ANs does not add up to be, or if that doesn't converge to zero, then the series is automatically divergent. In which case, I would then recommend to use the test of divergence. These limit calculations, the limit of the sequence for your typical calculus two student by the time you're this far in the series, these limit calculations are getting much, much, much easier. Many of these you can eyeball. And therefore, my recommendation would be that you could identify these limits pretty quickly. Therefore, use the test for divergence, right? I think that's one of the easiest ones to do. Some other ones that are super easy are looking for P series and geometric series. P series have a very simple convergence test. We take the sum where one, we take the sum where N equals one to infinity of one over N to the P, like so. And as such, we just have to look at the P value. If P is greater than one, the series will converge absolutely. If it's less than or equal to one, the series will diverge. And that's all there is for a P series. Geometric series are very similar. The geometric series, you're gonna get some type of exponential term popping up here, some R to some power of N. And the exact power of N doesn't matter too much because we can fluctuate it. We can adapt for it. And so if your ratio is small, that is if the absolute value of R is less than one, then the geometric series will converge absolutely. And if it's greater than or equal to one, it'll diverge. And so geometric series and P series are super simple tests to run. Now, another test I think that's relatively easy to use is the alternating series test. What do you have an alternating series? An alternating series will be specified because there's gonna be some type of like negative one to a power. You might have like negative one to the N, negative one to the N plus one, negative one to the N minus one, just as some examples. But you'll see this alternating factor of negative one in the series. This will resonate with you, oh, it's an alternating series. Therefore, the alternating series test might apply. The alternating series test is fairly nice to do in order to determine convergence, the series has to be decreasing and it has to converge towards zero. The sequence, I mean to say, the sequence, the positive sequence needs, the absolute sequence needs to be decreasing towards zero. And therefore, if those two conditions are satisfied, and again, this is often for calculus two students this late in the series, it's usually not too difficult to get a gut instinct about the convergence towards zero and decreasing and such. And the alternate series test that you mentioned can only show convergence. If the assumption of the alternating series test fails, I should mention that you can't get divergence using the alternating series test. Now, if the alternating series test fails, it's oftentimes this right here, the sequence doesn't converge towards zero, in which case if the sequence doesn't converge towards zero, then our good friend, the test for divergence to be used to show divergence. So that's good. One does have to watch out for that. If the function is not decreasing towards zero, then there's an issue there. But in order for this to work, we only need that it eventually converges towards zero. It has to be eventually decreasing towards zero. So if a function kind of jumps around a little bit and then kind of stabilizes towards zero, as it's decreasing, that's perfectly fine. The issue that one has to deal with is what if your sequence kind of like it goes up and then it goes down, then it goes up and then it goes down and then so it's like, it is dampering. It's kind of like a sine wave that's like dampering towards zero. That is sort of an issue one has to watch out for. The alternating series test wouldn't apply there. And based upon the function, we might want to use something like a comparison test, which actually is the next one on the list. The comparison test, if your series is comparable to something that is easier to determine it's convergent, then make that comparison. Who do we like to compare to? We like to compare to P series and geometric series. We're gonna do that all the stinking time. Now, one can use the comparison test, but honestly, I'm quite inclined to use the limit comparison test. Ironically, the comparison test has limits on what you can do. The limit comparison test is limitless because you can use limits to get past the limits of the comparison test. That wasn't confusing whatsoever, was it? The limit comparison test, I like it a lot because you can, the limit you actually have to compute associated to limit comparison test is again, usually not too difficult. We can often see it pretty quickly or maybe like a Lowby-Toll's argument could be used to help us out. On the other hand, making the inequalities work for the comparison test, in my opinion might be a little bit more difficult than doing the limits for the limit comparison test. So I'm more inclined to use the limit comparison test, but the comparison test could be great. And typically we want to compare to a geometric or P series, if at all possible. One thing I should mention about the comparison test and the limit comparison test is the terms in the sequence necessarily have to be positive. If you don't have positive terms, then you can't use either of the comparison test. But if you don't have positive terms, then you might have an alternating series test for which I would then tell you to go that direction. A test that's a little bit stronger but a little bit more difficult to do would be the ratio test or the root test. I'm putting the root test a little bit lower down the list because in my opinion, especially for students, the limit of a ratio is probably gonna be easier to do than the limit of a root. And so I'm inclined to suggest using the ratio test. The ratio test is gonna be great if you have products of things, exponentials, factorials. The ratio test works out really great. I would try to simplify the ratio. You take AN plus one over AN. You wanna then show does this thing, is this limit, I should say it approaches a limit, is this limit less than one? Is it greater than one? And if that's the case, you can use the ratio test. Now, if the limit was greater than one, it turns out it's gonna be divergent. You actually could have used the test for divergence, which if you tried that earlier, you would have gotten it already. Because the test of divergence is usually better than a negative ratio test. But it happens, it's okay. Remember though, if the ratio test is equal to one, then it's inconclusive and you'll have to use a different test. But if that's the case, again, the comparison test or alternating series test or something probably would have detected that previously. So if you're thinking about the ratio test, then you're probably at a point where it's likely this limit will be less than one, thus giving us convergence. But that's not a guarantee, right? That's just a recommendation, not a guarantee. The root test is very similar to the ratio test where you have to take the sequence of the nth root of the absolute value of an and look at this limit as it goes to L. The criteria is the same, greater than one is divergent. Equal to one is inconclusive, less than one is absolutely convergent. And so same hope there. But I would suggest using the root test when you have lots of like nth powers of things. But generally the ratio test will be a cleaner argument. And then kind of as the last resort, I'm inclined to use the integral test. When you have some type of continuous series, that is your sequence, f of n naturally extends to some continuous function f of x. The problem with the integral test is that all the other convergence test typically require the calculation of a limit or some type of inequality of some kind. The integral test, as the name suggests, requires we calculate an integral. That is we have to calculate an anti-derivative. And that can be very difficult to do. We have to sometimes, right? Sometimes the anti-derivative is pretty nice. And if I looked at a series and I thought the anti-derivative was obvious, it's like, oh, that's a super easy anti-derivative. Then I actually might jump to the integral test immediately. But as anti-derivatives are not typically intuitive, right? Using the integral test is basically turning a street fighting series problem into a street fighting integral problem. And that is probably not an easy thing to do. Now, there is one potential way out of this, right? Because we only care about the convergence of this thing. So the integral test says the convergence of the series is the same as the convergence of the integral. Now, improper integrals could be shown to be convergent or divergent using the comparison test that integrals had. But I wanna mention that if you're showing that this integral is convergent or divergent using a comparison, you might as well just show that the series was convergent using a comparison test. So what I'm trying to say is if you're gonna use the integral test, you're planning on taking an anti-derivative. There's basically no way around it. And I would again, hesitate to do that unless one, I think it's an easy anti-derivative or two, I've tried everything else and I'm still stuck. And so an anti-derived calculation might be my last resort. So those give us some general strategies on what you could do to determine the convergence of a series or the convergence or divergence of a series. And this kind of covers the series convergence test that we've learned about. And I've talked about the type of series that we might see in a course like this. So the next couple of videos will give you some examples. Try them out on your own and then compare them with my solutions. And if you do that, I think you'll learn a lot. So take a look for that in the link you should see right now.