 In this video, we'll discuss the convergence and divergence of infinite geometric series. The results are important for a number of reasons, the most critical of which, for our purposes, will become familiar later in this unit. As you read in the lesson, we know that the sum of finite geometric series with n plus 1 terms is given by this formula, where a is the initial term and r is the common ratio. What would it mean for us to investigate the sum of the associated infinite geometric series? Well, an infinite series would mean that our n approaches infinity. So, suppose we look at the limit of this sum as n approaches infinity. What we notice is that the limit of this expression as n approaches infinity depends on the value of r. We have four cases to consider. Suppose first that r equals 1. This gives us the geometric series as you see here. Since this represents the sum of an infinite number of the same constant, a, the sum is infinite. Therefore, this series would diverge. Now suppose that r equals negative 1. This gives us the geometric series as you see here. The partial sums of this series alternate between being 0 and a, but never reach a single finite value. Therefore, the series would diverge. Suppose now that the absolute value of r is greater than 1. In other words, the common ratio is less than negative 1 or greater than 1. In this case, we find that the limit as n approaches infinity of a times 1 minus r to the n plus 1 divided by 1 minus r is either positive or negative infinity. This is because the limit of this term r to the n plus 1 is either positive or negative infinity. So if the common ratio is greater than 1 or less than negative 1, the series would diverge as well. Now suppose that the absolute value of r is less than 1. In other words, our common ratio is between 1 and negative 1. In this case, we find that the limit as n approaches infinity of r to the n plus 1 equals 0. So our limit as n approaches infinity of the expression we've seen before is equal to a divided by 1 minus r, which is a finite value. So if the common ratio is between negative 1 and 1, we find that the series converges. But to summarize, an infinite geometric series converges if the absolute value of r is less than 1 and diverges for any other values r. Let's work through a couple of problems to apply our conclusion. Take this example and we're going to determine whether this series converges or diverges. We recognize that this is a geometric series with the initial term 5 divided by 4 to the 0, which is 5. The next term being 5 divided by 4 and so on. So our initial term is 5, so that's our A, and our common ratio is 1 fourth. We can see this because we can rewrite this series as 5 times 1 fourth the nth power. Now since r is less than 1, we know this series converges and even further we know that the sum is equal to 5 divided by 1 minus 1 fourth, which is 5 divided by 3 fourths, which gives us a sum of 20 thirds. As another example, we have the following series and I'm going to rewrite this to make it clear that this is a geometric series. I'm going to rewrite this as 9 to the n times 5 to the negative n minus 1. This then gives me that the series can be written as 9 to the n divided by 5 to the n minus 1, which is the same as the series from n equals 0 to infinity of 9 to the n divided by 5 to the n times 5. I'm going to rewrite this series this way and we see that our initial term is 5 and our common ratio is 9 fifths. Well, since the common ratio is greater than 1, we know that this series diverges.