 In this video, we're gonna discuss the solution to question 13 from the practice final exam for math 12-20. And in this series, or in this question, we're given the series, which is just some, or in ranges from one to infinity, of three n minus one over n plus one, and all of that, that whole fraction's raised to the nth power. We're supposed to first determine whether the series is absolutely convergent, conditionally convergent, or divergent. So you should indicate that such. So if you think it's absolutely convergent, you can just circle it on the page or just write down absolutely convergent, or whichever the other ones are. Now, it's not good enough just to answer, right? This is not a multiple choice question. This is a free response. So you do have to show your work in the process of doing this. And this, it means you include any convergence test you're using. And that can get kind of complicated at times, right? So if we were like to say, oh, this, I'm gonna limit comparison this to something else, right? Maybe that's sort of the approach you wanna take. Maybe you're saying that I'm gonna do a limit comparison of this thing to the sum where n goes from one to infinity, of say maybe three n over n raised to the n. You might do some type of limit comparison there. Well, I should see then what's the limit, right? Why are these things comparable using the limit comparison test? We should be looking at the limit of that thing, recognizing the limit goes to one. Then we're like, oh, this is convergent by such and such reason. So you would mention why you think this thing is convergent. And then you would mention the limit comparison test to say why that one's convergent. And that's just an option right here. And that might be a path that some of us take. For this one, I think an actually easier approach would be to use the root test for this one. This is sort of like the little brother of the ratio test to just tags along. And although it seems annoying for the most part, does turn out to be useful every once in a while, right? Especially when someone has to take the blame for something that the older brother did, right? So we wanna use the root test here. Because of this nth power for the entire expression, the root test works out really nicely. So for the root test, we need to consider the limit as n goes to infinity of the nth root of the absolute value of our sequence, three n minus one over n plus one, raised to the nth power right there. And so because of the nth root, when you combine with the nth power, they cancel out very nicely. So we have to take the limit as n goes to infinity of just the absolute value of three n minus one over n plus one. And so notice in this situation that as this is a balanced rational function, as this is a balanced rational function, this thing will converge towards three over one, particularly three. And so as this limit is greater than one, this tells us that then we can conclude that the series is divergent by the root test. And so that would be the answer we get if we use the root test as an approach. Now admittedly, if you don't want it to be so wordy, that's okay. If you just said something like, oh look, we took the limit and we timed out the limits greater than one, cite the root test, we use Minjin's name, that's okay. But then you should indicate that, okay, it's divergent. Some somehow or another, I should know that you believe the series is divergent, that the reason you believe it's divergent is because you used the root test. And since you're using the root test, you have to show me the limit calculation associated to the limit test or the root test, which is right here. If you're using the ratio test, I wanna see the limit calculation involved with the ratio test. If you're using limit comparison test, right, what's the limit calculation of using the comparison test? What are you comparing to? So you need to provide the evidence that's not just mentioned the test, but what are the details? What are the assumptions? What's the criteria necessary to use that test? Now this one turns out to be pretty easy. I feel if you recognize the root test as the right tool, the ratio test works out okay, it is a little bit more messy than the root test. The root test works perfectly here. But this question number 13 here, this is a street fighting type question. It could be anything. It could use the root test like this one. It could use the ratio test, the integral test, right? It could be a geometric series for all we know. And so be able to prove the convergence or divergence of any of these things. It will distinguish between absolute convergence and conditional convergence. And so do be aware of that. Absolute convergence is actually usually what we can see when it turns out to be convergent because like with this sequence right here, everything, all this terms in the sequence are positive. So it's convergence, it's gonna have to be absolutely convergent. So I would wanna caution the viewer right here is, can you prove that something is conditionally convergent? And I actually have a video of such an example linked on the screen right now because to show that something's conditionally convergent, it would require that you show that it's convergent oftentimes using the alternate series test, but it's also not absolutely convergent. So there are two important ingredients and a mission of one would be some foreverture of some points there.