 Welcome to this quick recap of section 8.2 on geometric series. Geometric series are a particular kind of sum that involves numbers with very nice patterns. First we'll take a look at geometric sums. A geometric sum with a notation s sub n is a sum of numbers in a certain form written here. The thing to notice about this sum is that we begin with sum number called a, and then every term afterwards is the same as the one before it, multiplied by one higher power of this number r. This number r is called the common ratio of the geometric sum. Here's some examples of geometric sums. In each of them notice how there's a number raised to steadily higher powers, the common ratio, and there is the same number out front of each of these terms, the number a. Geometric sums can represent a variety of situations. For example, a bank account earning interest can have an additional amount added on each month that is a certain fraction of what was already there. We're often interested in how to sum up the final value of such a situation. In this case, instead of adding up the geometric sum term by term, we can use this shortcut formula. So a formula for the geometric sum s sub n is given by this formula, which involves the beginning amount called a, the common ratio called r, and the number of terms called n. This formula only applies if the common ratio is not equal to one. If it were equal to one, we would be dividing by zero. If you're interested in knowing what happens when r is equal to one, try substituting it into the left-hand formula here, and you'll see that we're adding up the number a over and over. A geometric series is simply an infinite geometric sum, meaning a geometric sum that never stops. We can write it either as we do on the right, which is a geometric sum plus dot dot dot, meaning continuing in the same pattern where each term has one higher power of r than the previous one. Or we can write it as on the left using summation notation. This is the same sort of notation that we used for Riemann sums in Calc I, and it represents the same infinite sum. Again, a is a real number, and this number r is called the common ratio. Here are some examples of geometric series. These are similar to the ones we saw on the previous slide, but they go on forever. To test your comprehension, you should make sure that you understand how each infinite series on the right is represented using the summation on the left. A little bit of notation. First, if we stop geometric series after the n terms, we get what we call the nth partial sum of the geometric series. This is the exact same thing as a geometric sum because a geometric series is a geometric sum that keeps going. In addition, we have a shortcut for summing up a geometric series value. It might seem counterintuitive that a geometric series with an infinite number of terms can actually have a finite value, but it is possible. This is very similar to what happened with improper integrals in Chapter 6. So if the absolute value of the common ratio is less than 1, then a geometric series sums to a finite value given by this formula. It's worth noticing that this formula is only valid if the absolute value of r is less than 1, that is, if r is between negative 1 and 1. Even though you might be able to evaluate this formula when r is outside that range, the results are invalid and make no numerical sense. However, if r is within this range, you can sum the infinite series. Finally, here's a summary of the similarities and differences between geometric sums and geometric series. Now that we've seen these, let's take a look at how to use them.