 Hello, and welcome to this screencast. In this video, we will calculate the sum of an infinite geometric series. The series we will consider is the infinite geometric series, the sum from n equals 1 to infinity of negative 3 to the nth power, divided by 4 to the n plus first power. This is a series because we are adding the terms of a sequence where the general term is negative 3 to the n, divided by 4 to the n plus 1. And we call this series geometric because, as we will see, consecutive terms have a common ratio. And it's an infinite series because we are adding up infinitely many terms. So let's start with some information to help us calculate our sum. If an infinite geometric series has a first term a, a common ratio r, and the absolute value of r is less than 1, then the sum of that series is a divided by 1 minus r. And you can check your textbook for the derivation of this formula. So our job is to find values for the variables a and r, and then we want to see if we can use this formula to calculate the sum. So let's start by writing out the first few terms of our series. When n is equal to 1, the numerator of our first term is negative 3 to the first power, and the denominator is 4 squared. When n is equal to 2, the numerator is negative 3 squared, and the denominator is 4 cubed. Our following terms that we're adding are negative 3 cubed divided by 4 to the fourth, and negative 3 to the fourth, divided by 4 to the fifth, and this pattern is going to continue on. So we're going to rewrite these terms, and that's going to help us find values for the variables a and r. So our first term, we're simply going to rewrite this as negative 3 divided by 16, and then we can start by writing the next term as negative 316, but then we're going to multiply it by negative 3 fourths. So altogether there are two factors of negative 3 in that numerator and three factors of 4 in the denominator, so that is equivalent to what we have written above. Our next term can be written as negative 316, but instead of writing negative 3 fourths, we're going to multiply by negative 3 fourths squared. And our following term is negative 316 times negative 3 fourths cubed, and this pattern is also going to continue in our terms. Since a represents the first term of our series, then that's going to be very easy to identify as simply negative 3 sixteenths. The variable r represents the common ratio of the series, and we note that when we list the terms that we're adding up, we can see that each new term is equal to the previous term multiplied by one more factor of negative 3 fourths. So we saw negative 3 fourths, negative 3 fourths squared, negative 3 fourths cubed, each time we have one more factor of that negative 3 fourths. And so that factor is the common ratio of our series. And since the absolute value of negative 3 fourths is strictly less than one, that satisfies the hypothesis of our information above, and so we can use that formula. So we have the sum of the series is equal to a divided by one minus r, and we can substitute our values for a and r. So the numerator becomes negative 3 sixteenths. And in our denominator, we have one minus r, which is one minus negative 3 fourths. We want to keep track of that negative. So in our next step, we're just going to simplify that denominator. So one minus negative 3 fourths is equal to positive 7 fourths. And now we have a fraction divided by a fraction. So we know that dividing by a fraction can be rewritten as multiplying by the reciprocator reciprocal of the fraction in the denominator. So now we have negative 3 sixteenths times four sevenths. We can cancel a factor of four from both the numerator and the denominator. And then our last job is just multiply this out and we get negative 3 divided by 28. So even though we're adding up infinitely many terms in our series, the result of that sum is a finite number negative 3 twenty eighths. And that's our final result. Thanks for watching.