 Come to this quick recap of section 8.4 on alternating series. An alternating series is a special series that has a particular form. The form is represented by the summation below, but the key idea is that an alternating series has terms that alternate between positive and negative. That's written at the bottom here. We either start out with a positive term, then a negative one, then positive, negative, and so on, or we can start with a negative sign as long as we're always alternating back and forth between positive and negative. This is represented in the summations by putting a negative one to a power out front. As the power increases, this flips back and forth between a positive sign and a negative sign. In order to do this, we've assumed that these a's of k's are greater than or equal to zero. That means that when we put a negative sign in front of them, they really are negative and vice versa. The sequence of a's of k's is called the sequence of positive terms. It's the sequence we get when we ignore the negative signs in front of the terms of an alternating series. An example of an alternating series that you've already seen is any geometric series with a negative ratio, as you can see below. In this example, the sequence of positive terms is listed below, and it's the same list of numbers, but positive. Remember, an alternating series is a series, so we are summing up a bunch of numbers, whereas the sequence of positive terms is a sequence, just a list of numbers. One of the main reasons to study alternating series is the alternating series test. This is a test which tells us exactly when an alternating series converges or diverges. If we have an alternating series as written here, then it converges if and only if the sequence of positive terms decreases and goes to zero, as k goes to infinity. In other words, the sequence of positive terms must converge to zero on its own, and it must do so by always decreasing. We can visualize why this is true by looking at the partial sum, s of n, where we add up the first n terms of the alternating series. Our first term lies somewhere on the number line. When we get our second partial sum, in which we take the second term of the series and add it on, we've actually subtracted this because the series is alternating. That puts us somewhere lower on the number line. The next term must be positive because we're alternating, but we've also assumed that each term is a little bit smaller than the one before it. That's why we're assuming that the series has terms a of k that decrease to zero. Because of this, the next term in this series can't possibly go above the first term that we looked at, and so we're starting to get trapped getting closer and closer to some limit in the middle. The next term must be alternating, so it takes us down the number line a bit, but again, we subtract a little less than we have before. And as we continue on, we spiral inward more and more until we eventually must become trapped and end up at some limiting value. That means that the series has converged to that value. This is a very powerful test because it tells us exactly how an alternating series acts. If we meet these hypotheses or conditions, the series is guaranteed to converge, and it's also guaranteed to diverge in any other circumstances. An important warning, this test only works on alternating series. As soon as we don't have an alternating series anymore, the test tells us nothing, it doesn't apply at all. The alternating series test leads us to the alternating series estimation theorem. If we have an alternating series that converges to sum number s, then the nth partial sum, s of n, satisfies this inequality. The way to read this is to say that the distance between the real sum and the partial sum is less than or equal to the next term in the series, the one that we stopped adding at, or that we would have added next if we hadn't stopped at term n. We can see why this is true by again looking at the spiral where we add up subsequent terms in the series. After a few sums, we know that the sum, the ultimate sum of the series, must end up in this highlighted region somewhere because we can never escape from that as we add more and more terms on. That means that the distance between the real sum, which might be around here, and the last partial sum that we've looked at, can't be any bigger than the next term, a of k in the series, because that term can't be any bigger than the yellow highlighted area. This tells us that the n plus 1th term gives us an estimate for how close our partial sum is to the real sum, and it tells us how to get as close as we want to the exact sum. Again, as a warning, this only works for alternating series, not for any other type of series. Finally, we'll take a look at a related type of series. If the series a of k is any series, that is, it's not necessarily alternating. It could have both positive and negative terms in any pattern. Then we can look at a related series in which we put absolute values on every term. We can think of this as the series where we make every term in the series positive, so every term is as large as it can be, that is, they're all positive. An example of this is shown here, where the left series has some negative and some positive terms, although not necessarily in any pattern, and the right hand series has the same terms but all positive. It turns out that this series with absolute values is important in terms of determining convergence of related series. Here's two new words that we should know. The original series, the series a of k, converges absolutely, or is absolutely convergent, if the series with absolute values also converges. On the other hand, the original series, a of k, converges conditionally, or is conditionally convergent, if the series of absolute values diverges, but the series without absolute values converges. The lesson from this is that changing some of the signs in a series can have a big effect on convergence, but in particular, with absolute convergence, if we make a series entirely positive and it still converges, then any way that we put negative signs into the series, the series will still converge. This is the power of absolute convergence. It tells us that many related series all converge. Now that we've seen these, let's take a look at how to use these tests and ideas in practice.