 Hello. Welcome to this quick recap of section 8.3, series of real numbers. In this section, we meet the idea of an infinite series, which is the sum of the entries in an infinite sequence. If we start with an infinite sequence, like a of n written below, and add up each of the terms, we get an infinite series, which can be represented by the summation on the right. We can also write it in two other ways written below here, all of which represent the same infinite sum. We've already seen an example of this in section 8.2 where we met geometric series. These are a special type of infinite series in which the numbers follow a special pattern. Whenever we have an infinite series, we can have the nth partial sum of the series. This is the finite sum, called s of n, in which we add up the first n terms of the series only, as written on the right. This is guaranteed to be a finite number because we're adding up a finite number of real numbers. When we list these, s of 1 being the first term, s of 2 being the sum of the first two terms, s of 3 being the sum of the first three terms, and so on, these form a sequence in their own right, called a sequence of partial sums. We usually call this s of n with a capital S. This terminology can be confusing, so we're going to go through it again in slow motion. We start with a sequence called a of k, and then we sum the terms in a of k to get a series. Then we make a new sequence, called the sequence of partial sums, and listed by a different name, s of n. This consists of the first term in the series, then the sum of the first two terms in the series, the sum of the first three terms in the series, and so on. So we're basically looking at where the series is going as we add more and more terms onto it. The nth partial sum is important because the sequence of partial sums is how we decide if a series converges to a real number or diverges. The series is said to converge if its sequence of partial sums converges, otherwise the series diverges. In other words, the series, a k, is described in the same way as the limit of its partial sums. We'll take a look at this again in a future video. Next we're going to take a very brief look at some tests to determine if a series converges or diverges. These tests only apply to certain types of series, but they help us determine if they converge or diverge. In general, other than looking at the sequence of partial sums, there's no guaranteed way to decide if a series converges or diverges. These tests will help us in certain circumstances and act as a sort of shortcut. Each test has three important parts. It has a setup, which is some initial work or calculation that we should do before using the test. It has hypotheses, also called conditions. These are labeled with the word if. The test only applies if the series that we're trying to test satisfies these hypotheses or meets the conditions. Finally, there are conditions listed with the word then. This is what we're allowed to conclude about the series, but only if the conditions have been met. One more warning. If a test's hypotheses, that is conditions, are not satisfied, then we can't say anything about the conclusion of the test. It could be anything. The series could converge or diverge, depending on the scenario, and we'll have to find a different way to decide that. With that in mind, let's look at our first test, the divergence test. In this case, if the limit of the ak's is not equal to zero, then the series where we sum the ak's diverges. Remember here that the sequence ak is the original sequence that we're adding together. This is not the same thing as the sequence of partial sums. An important warning. If the limit of the ak's is zero, then we cannot draw any conclusion. There are series that converge when ak goes to zero, but there's also series that diverge when ak goes to zero. The image below gives an idea of why this test is true. Each red dot is one term in the sequence ak. They seem to be going towards one-half. If we imagine adding together the values of each of these terms, we're always going to be adding about one-half into the series. If we add enough one-halfs, we can get larger than any value we would like to choose, meaning that the series will diverge. Our next test is the integral test. In this one, we need a function f, which is decreasing and positive as long as x is bigger than some number, c. And we need a of k to equal f of k for each positive integer k. In other words, we need a function whose values match the values of the sequence that we're going to add up. If the improper integral from c to infinity of f of x converges, then the series in which we add up the ace of k's also converges. On the other hand, if that same improper integral diverges, then the series diverges. In order to use this, we have to find a function that matches the values of ace of k, and the function must be eventually decreasing and positive. That is, it's decreasing and positive as long as we're to the right of some x value. The image is below gives an idea of why this is true. The area under a function, shown by a red line, is approximated by these boxes where the height of each box just touches the curve. That's the value of the individual term in the series that we're adding up. These boxes approximate the function, and it turns out that the improper integral either converges or diverges in exactly the same way that the series converges or diverges. Our next test, the limit comparison test, requires two series called ak and bk. These series must have positive terms. For our hypotheses, if we calculate the limit as k goes to infinity of b of k over ace of k, that is, a general term in series b over a general term in series a, and we get a number c, which is positive and finite, then the two series, ak and bk, either both converge or they both diverge. In other words, they have the same behavior. This test only works for positive series, not for ones that ever have a negative term in them. And we typically apply the limit comparison test when we already know the behavior of one of these two series. That way, once we calculate the limit, we can then determine exactly what the other series does. Finally, the ratio test is like a limit comparison test where we compare a series to itself. We calculate the limit as k goes to infinity of the absolute value of the k plus one-th term in a series divided by the k-th term in a series, also with absolute values. If this comes out to a real number called r, then one of three things can happen. If this number is between 0 and 1, including 0 but not including 1, then the series converges. If the number is bigger than 1, then the series diverges. And finally, if the number is equal to 1, we cannot draw any conclusions. We have to find a different test to tell us what happens. The absolute values matter here. It keeps our ratio positive, which is important. This test effectively compares a series to itself. And if r equals 1, we truly do not get any information. We cannot draw any conclusions. Now that we've seen all these, let's take a look at how to use each of these and how to look at infinite series and their sequence of partial sums in more detail.