 In this video, we're going to discuss the solution to question 5 from the final exam for Math 1220. In this question, we're asked to determine which of the following series is convergent by the alternating series test. So notice the test it specified. It says the alternating series test. So whether we have a convergent series or not, we only care about the ones that can be proven convergent by alternating series. And by the alternating series test. And as such, if a series is not an alternating series, then the alternating series test does not apply. That rules out very quickly choices A, B and F, which are not alternating series. You don't see this negative 1 to the n or this negative 1 to the n plus 1 that's necessary for our alternating series. Now be noticing here that choice B is a p-series, right? This is where p equals 2. And so as a p-series, it is convergent, but we're not looking for a convergent series. We're looking for a series which is convergent by the alternating series test. So the instructions matter here a lot. Now for a series to be convergent by the alternating series test, it's necessary that if you ignore the negative 1 to the n, that that sequence decreases towards 0 eventually, right? And so looking at these, once we get n squared over 3n squared plus 1, that's a balanced rational function. If you just look at the leading terms, you get n squared over 3n squared, right? That sequence, not the series, but that sequence would converge towards n squared over 3n squared. That is one-third. So this sequence is in fact divergent by the divergence test. So that's not the correct answer right there. Working over here on E, we actually see something very similar. As n goes to infinity, again ignoring the negative 1 to the n right here, just the sequence 1 over the arc tangent of n, as n goes to infinity, you're going to end up with 1 over the, well, the horizontal asymptote of arc tangent, which is pi over 2, that becomes 2 over pi. But in particular, that thing is not 0. Therefore, this is a divergent series by the test of divergence. So it would appear that by process of elimination, D is the only correct answer, but we might have actually found this earlier. Again, ignoring the negative 1 to the n plus 1, if we look at top and bottom, this sequence is essentially the square root of n over n, which is to say 1 over n to the 1 half. And so as n goes to infinity, you get 1 over infinity, which will look like 0. So this is a decreasing sequence, which approaches 0. It's an alternate series. Therefore, it will be convergent by the alternate series test. We might have actually found D a little bit sooner. So you don't necessarily have to go through each and every one, but that act might be a good measure, right? Just to make sure you get the right answer on this question, you could eliminate every possibility, except for the one correct one that does work. So this one requires the alternate series test, which apparently only applies to alternate series. So watch out for that on this question.