 Welcome to this screencast on section 8.3 using the ratio test. This is a particularly important test for the rest of this chapter, so we're going to spend some time on it. Here's a reminder of what the ratio test says. If we have an infinite series, ak, then we calculate this limit. The thing to notice is this limit is the ratio of the subsequent term of a series divided by the current term. This is the same sort of calculation we would do to find the common ratio in a geometric series. But in this case, we take a limit as k goes to infinity. And if we get a number, r, or if we get infinity, then we can use that to determine what the series does. If r is between 0 and 1, including 0 but not including 1, then the series converges. And in fact, it converges absolutely, which is something we'll learn about in the future section. If r is bigger than 1, and that includes infinity, then the series diverges. But if we get r is exactly equal to 1, then the test is inconclusive and we have to try another test. The ratio test is useful because it treats a series as if it were geometric, and it calculates what the common ratio would be. So of course, we can use this on series that aren't geometric. But what the test does is it takes the ratio of two terms, and by taking the limit as k goes to infinity, it calculates what this ratio eventually comes out to be as the terms of the series go along. And then we can use almost exactly the same numbers as we would with a geometric series. This test is often used with exponents and factorials since they'll cancel in the fraction that we have to calculate. So let's get on to an example here. Our first example is to determine the convergence of this series. Well, this series has both an exponent and a factorial, so it's a great candidate for the ratio test. And that's probably the first one I would try. So I have my limit already set up here. In the numerator, I need the k plus 1th term. So that just means every k will turn into a k plus 1, which is not too difficult to calculate. So I have a 3 to the k plus 1 over, and for the k factorial that turns into a k plus 1 factorial, notice the parentheses. I'm taking the factorial of the whole number, k plus 1. My denominator is the kth term, so that just looks like this. We often end up using the ratio test with fractions, and so it's good to simplify those fractions quite a bit. The way I'll do it here is I'll multiply by the reciprocal of the bottom to get me just a single fraction. So I'll start out with the same numerator, and then I'm multiplying by the reciprocal of the denominator. And I can see right away that a lot's going to cancel. My 3 to the k's are going to cancel, leaving just a 3. So I'll do that step first. I'll have a 3 here, and also a k factorial left over. And I'll be dividing by a k plus 1 factorial still. All the 3's in the denominator have canceled. Now we're going to take a look at a really common maneuver with this sort of thing, which is to simplify factorials. So here's my limit one more time. I'm going to write out what the factorial actually means. So I have a 3 out front from the powers. Now I'm going to write the k factorial. That's 1 times 2 times 3 and so on, all the way up to k. And in the denominator where I have a k plus 1 factorial, well that also means 1 times 2 times 3 and so on. Up to k, and then it goes one more. It goes up to k plus 1. Putting back in the absolute values, which do matter here, as we'll see in the future, I can see that a lot of this is going to cancel. In fact, everything in the factorials will cancel, except for that k plus 1. And so I'm left with this limit. I left with a 3 in the numerator, and a k plus 1 in the denominator. And that comes out to 0. That's good, because one of the numbers I can use with the ratio test is 0, and that says that this series converges. Think of this as exactly the same sort of conclusion you'd get with the geometric series test. So in this case, I can write something sort of like this. I can say, therefore, this series converges by the ratio test, and it would be nice to add because r equals 0, or the common ratio after this limit comes out to 0. So here's our classic example of how to use a ratio test. We'll do two more examples that show slightly different circumstances. So here's our next example. Determine the convergence of this series. So notice I've changed my index to n. That's fine, my index can be anything as long as I'm careful with it. So I'm going to write out that common ratio again. Now I will point out the ratio test might not be your first choice for this series. And so you might actually want to pause and think what would be a good choice of a test for this series. All right, so I'm going to write out what the ratio test would give us because I want to demonstrate the ratio test here. And if I do this, I can combine these very quickly and see that I'm going to get an n over an n plus 1, and that limit comes out to 1. So this says that the ratio test is inconclusive, which just means that we don't know. Doesn't mean that we actually know any kind of conclusion. It means we'd have to go to another test. So you might remember there's lots of other options for this one. For example, we could use the integral test, or we could just remember that this is the harmonic series, and we've seen several reasons why this diverges. So we could actually say, for example, that this is a p-series, if you remember those, from the integral test, with p equals 1, so it diverges. So if you get a ratio of 1 from the ratio test, you need to find another test and use that. All right, let's take a look at one last one here. Here's another example of a series. For this one, it's reasonable to use the ratio test, but again, you should pause and see if you can think of a simpler way to deal with this series. All right, we are going to use the ratio test to demonstrate what it does. So I'm going to write out the k plus 1th term here. All the k's become k plus 1s, which does involve a lot of writing, but this isn't too difficult to do. And again, because we have a fraction like this, a lot is going to simplify, and that's a good reason to use the ratio test. So again, I'm going to multiply by the reciprocal, but before I do that, I'm going to notice that these absolute values will just kill off the negatives here. They're gone entirely, and so I'm going to remove those. It's not that they cancel, it's that the absolute values remove them. So when I write equals here, it's equal because the absolute values will get rid of negatives. All right, so in the numerator, I still have an e to the k plus 1, and the denominator, I still have a 2 to the k plus 1. Multiplying by the reciprocal, I'm going to get a 2 to the k and an e to the k. Now you can see that a whole lot of things are going to cancel here. Those e's will go, the 2's will go, and I will be left with just the limit as k goes to infinity of e over 2. And if you remember that e is a little bigger than 2, that's a number which is a little bigger than 1. So that says that this series diverges using the ratio test. So you could write that this way, therefore this series diverges by the ratio test. You might have noticed that this actually is a geometric series. So we could have used just the geometric series test and noticed that the common ratio was actually negative e over 2, and that's again outside of the range from negative 1 to 1 where the geometric series automatically converges. So this is a good example of why the ratio test really does act like a geometric series test. It treats a series just like a geometric series, and if you have a geometric series, you'll get the common ratio with absolute values on it. And if you don't have a geometric series, you'll get something related to it that shows what ratio your terms eventually get.