Hi, and welcome. My name is Anthony Varella. And today, we're going to find the sum of an infinite geometric sequence. So we'll start by talking about divergent and convergent sequences, and then talk about their sums, which we call series. And we'll use a formula to find the sum of an infinite geometric sequence.
So let's take a look at this geometric sequence. Here we have a common ratio of 2. So we multiply 2 by 2 to get 4, multiply 4 by 2 to get 8, and multiply 8 by 2 to get 16. And this is an infinite sequence. We are continuously multiplying a term by 2 to get the value of the next term. Now, we can also have divergent sequences with certain common ratios that are negative. So although our terms are alternating between positive and negative, they are divergent-- they are going off towards the infinities. So with divergent sequences, the absolute value of the common ratio is greater than 1. So greater than 1 or less than negative 1. And with divergent sequences, if we were to add the terms in an infinite divergent sequence, the series or the sum would be divergent, as well, because the terms themselves don't have any limit. So the sum doesn't have any limit. It'll be divergent to either positive or negative infinity.
Well, now let's compare this, then, with convergent sequences. So here is a convergence sequence. Our common ratio is 0.5. So we're always cutting the term in half to get to the value of the next term. So we have 32, 16, 8, 4. So on and so forth. Now we can also have a convergent sequence with certain common ratios that are negative. So here's a common ratio of negative 0.5. And once again, although our terms are alternating between positive and negative, the value of the terms do converge towards zero, because the next term is always going to be a portion of the term before it.
So when we're thinking about convergent sequences, the absolute value of the common ratio is less than 1. So that would be in between negative 1 and positive 1. And when we're adding the terms in an infinite convergent sequence, so that series or sum is also convergent, because we are going to be adding a value that converges. So eventually, then, our sum is going to in theory converge to a certain value as we add an infinite number of terms.
So when we're thinking about the sum of a finite geometric sequence, a sequence that has a certain number of terms in it, we can use this formula that the sum of n number of terms equals the value of the first term multiplied by 1 minus r to the nth power over 1 minus r, where r is the common ratio and n would be the number of terms. Now, in an infinite geometric sequence, then, n is infinity. Or it's infinitely large.
So how do we make sense of this formula, then, when n is infinity? Well, with divergent sequences, that means that r raised to the power of n is going to diverge to the infinities. So the sum is also divergent. So we can't really use this formula. We know that the sum of an infinite divergent sequence is infinity.
So we're going to consider, then, convergent sequences. So our common ratio is in between negative 1 and positive 1. So when n is infinitely large, r to the n converges to 0. So 1 minus 0 is just 1. So our formula for an infinite geometric sequence is simplified a bit. We just have 1 over 1 minus r being multiplied by the value of the first term. And this would only apply, then, to our convergent sequences, because the series, or that sum, converges to a certain value.
So let's use, then, this formula to find the sum of an infinite geometric sequence. So here is the infinite geometric sequence that we'd like to sum. We have 48, negative 24, positive 12, negative 6, and this is an infinite sequence. Well, we know it is a convergence sequence because our common ratio, the absolute value of the common ratio, is less than 1. The terms are going to converge towards zero, because we're continuously dividing by 2, and then changing the sign, because we can see that our common ratio is a negative number.
So we are going to then use our formula. We know the value of the first term is 48. So now we need to write in our 1 over 1 minus r. So what's the common ratio r? Well, to get from 48 to negative 24, we multiply by negative 1/2. To get from negative 24 to positive 12, we multiply it by negative 1/2. And to get from positive 12 to negative 6, we multiply it by negative 1/2. So our common ratio is negative 0.5.
So now let's simplify what we have as the fraction 1 over 1 minus negative 0.5. That would be 1 over 1 plus positive 0.5. So we have 48 times 1 over 1 and 1/2. Well, let's cleanup our 1 over 1 and 1/2. That would be 3 over 2. And when we multiply that by 48, we get a converging sum of 32. So the sum of this infinite sequence, then, converges to a value of 32.
Let's review our lesson on the sum of an infinite geometric sequence. With divergent sequences, the common ratio, the absolute value of the common ratio is greater than 1. So the sum of an infinite divergent sequence is also divergent. With convergent sequences, the absolute value of that common ratio is less than 1. So if we were to add up all of the terms in an infinite convergent sequence, that also converges to a certain value. And to find that value, we use this formula, the sum of an infinite geometric sequence that is convergent is the value of the first term multiplied by 1 over 1 minus r, where r is that common ratio between the terms.
Thanks for watching this tutorial on the sum of an infinite geometric sequence. Hope to see you next time.