 Welcome to this screencast on section 8.4 where we'll talk about conditional convergence. So here's our problem. Does this series converge absolutely, converge conditionally, or diverge? Anytime I'm trying to figure this out, the first thing that I'll do is just try to figure out whether the series converges or diverges, and then deal with absolute or conditional convergence if we've found that the series converges, otherwise we don't have to worry about it. So let's start by figuring out convergence. The series should look somewhat familiar to you, so the first thing I want you to do is pause the video and decide what kind of a series test or convergence test would apply in this situation. What's the best one to use? All right, we're back. I would recommend that we try to use the alternating series test. We can see from this part right here in the series that this is going to be alternating, and so the alternating series test is a very nice choice that automatically tells us exactly what happens. So the first thing I should say is that this series is alternating, and anything more that we want to do, we need to know the sequence of positive terms, which we usually call a sub n. So I'm going to name that with a sub n is 1 over squared of n. That's the positive part of the terms in this series. Everything else we want to do depends on what behavior the sequence a sub n has. So we need to know about two things. First, does the sequence a sub n decrease, and second, does it converge to zero? And if we can answer yes to both of those, the alternating series test will tell us that this converges, and otherwise it will tell us that it diverges. So first, we can definitely say that the sequence a sub n is decreasing. Why do we know that? Well, because if I write out the term a sub n, I'll label that here just to remind us what it is, and I also write out the next term, a sub n plus 1, which would be square root of, or 1 over square root of n plus 1, then I can automatically see that the fraction on the right has a bigger denominator, and when I'm dividing by a larger value, I get a smaller term. So that means I can put the inequality this way. So that means that I've always shown that as we go further along in the sequence, the terms get smaller and smaller because I'm dividing by larger and larger numbers. So we have a decreasing sequence. The next question is, what's the limit as n goes to infinity of the terms in this sequence of positive terms? Well, that's the same as the limit as n goes to infinity of 1 over square root of n, but I know what this does because I know what the limit of the related function, 1 over square root of x is as x goes to infinity, and that's 0 because I know the limit of the related convergent function. So because of that, I have a sequence that's decreasing and the limit of it is 0, and that means that the alternating series test tells me that this series converges. So I would write it this way, by the alternating series test, the series we have here converges. So we've decided that we're in one of these two circumstances, but we don't know which of those two happens, so we need to do some additional work. So we'll do the additional work right here. In order to decide whether the series converges absolutely or converges conditionally, we need to look at the series where we put absolute values on the general term of the series. The absolute values aren't too difficult to deal with here because all they do is get rid of the negative signs. The only negatives that we have in this are the numerator where we have negative 1 to the n, and so I'm left with the positive terms 1 over square root of n. And if I wanted to, I could rewrite this in the form n equals 1 to infinity, 1 over n to the one-half, since the one-half power is the same as the square root. So right now, we want to figure out if this series converges or diverges. So please pause the video for a second and decide how you would determine the convergence of this new series. All right, we're back again. I would say that this is a p-series if you remember what those are, and in fact, this diverges because it's a p-series and not only a p-series, but it's a p-series with p equals one-half. And that's the key thing for p-series. If your p-value, your exponent, is less than or equal to 1, you automatically diverge, otherwise you converge. You could also use the integral test in this case, do an improper integral of 1 over square root of x from 1 to infinity, and when you calculate that, you'd find that it diverges also. So now that we know that, that tells us exactly what happens to this sequence or to this series, excuse me. And that's because we've determined that the original series converges, but now this new series with absolute values diverges. And so that means, and I would write it this way, that the series given to us converges conditionally because the original series converges as we showed on the last slide, but the series with absolute values diverges. If the series with absolute values had converged, we would have said it converges absolutely, but in this case, we have that it converges conditionally. A conditionally convergent series is a very strange object. This means that it almost diverges, it just barely converges. In fact, one of the stranger things you can do with a conditionally convergent series is by rearranging terms, you can get it to sum up to any number that you want. In particular, that should tell you to be very careful when doing any kind of algebra on a conditionally convergent series because very strange things can happen.