 In this video, we provide the solution to question number seven for practice exam number four for math 1220, in which case we're given six infinite series that you can see right here. And we're asked to determine which of the following series is convergent by the alternating series test. Now be aware that alternating series test only applies to alternating series that is series that switch between positive and negative. So I can rule out many of these because they're not even alternating series. So whether they're convergent or not is doesn't matter. Like this is a convergent series, but it's not convergent by the alternating series test because it's not even an alternating series. So that itself also eliminates half of the options here. So we have three alternating series like so. Now to apply the alternating series test, you have to look at the absolute sequence. Does that thing decrease? Well, so it's got to be positive, which it is because we're ignoring the alternating sign now. Is it decreasing towards zero? It has to eventually be decreasing towards zero. And now if you look at C there, when you look at the sequence N squared on top, three N squared on the bottom, that sequence as you take the limit, the sequence, and again, I'm not talking about the series, I'm talking about the sequence there. The sequence is going to go off towards, we'll write this out here, the sequence is going to be going off towards one third in that situation, that's not zero. So since that sequence is not going towards zero, it's not convergent by the alternating series test. It's actually divergent by the divergence test. So we're going to throw that one out from any further consideration here. With this one right here, if you ignore the alternating factor, is that thing decreasing towards zero? Well, the top is approximately the square root of N, the bottom is N itself. So this thing is approximately the same thing as the sequence. I'm not talking about the series at the moment, we have to analyze the sequence and know the convergence of the series. So this thing is approximately one over the square root of N. And as N goes to infinity, that does go towards zero. There could be concerns about, is it always decreasing? Well, it doesn't have to always be decreasing, it has to be eventually decreasing. So there actually is some promise with D there. I feel like that's the right answer. The sequence is an alternate sequence that converges towards zero in a decreasing manner. At least it seems like it probably should be decreasing. I have some hesitation there. So let's look at the last possibility just to make sure. Because again, you could show that it's decreasing. You can. If you look at the other one here, I want to show this one doesn't work. Because if you take one over the arc tangent of N, as N goes towards infinity, one over arc tangent is actually going to go off towards one over pi halves, which is two over pi, which is not zero. So again, this one doesn't diverge by the alternate series test. It actually diverges by the divergence test. So by process elimination, it has to be D. But it also agrees with what we saw earlier that, yeah, the sequence does converge towards zero. It's decreasing towards zero. At least eventually it's decreasing towards zero. So the correct answer would be D.