Lesson Video: Sum of an Infinite Geometric Sequence Mathematics

Video Transcript

In this video, we’ll learn how to calculate the sum of an infinite geometric sequence. Let’s begin by recalling what we know about geometric sequences.

A geometric sequence is one which has a common ratio between consecutive terms. In these cases, to get from one term to the next, we multiply by that common ratio. And we can find the value of our common ratio by therefore dividing any term by the term that precedes it. Now, for a geometric sequence with a first term of 𝑎 and a common ratio 𝑟, the 𝑛th term is given by 𝑢 sub 𝑛 equals 𝑎 times 𝑟 to the power of 𝑛 minus one. But we might also recall that we can find the sum of a finite geometric sequence. And when we sum the terms in a sequence, we call that a series.

For a geometric series with first term 𝑎 and common ratio 𝑟, the sum of the first 𝑛 terms is 𝑎 times one minus 𝑟 to the 𝑛th power over one minus 𝑟. It’s also worth remembering that this is equivalent to 𝑎 times 𝑟 to the 𝑛th power minus one over 𝑟 minus one. Now, it isn’t necessary to switch between these two forms, but it can be useful if we need to perform calculations by hand. When 𝑟 is greater than one, we tend to use this form. And if 𝑟 — the common ratio — is less than one, we tend to use this form.

Now, the formulae we’ve given so far are really useful for solving problems involving geometric series. But what we’re interested in is looking at how we might use these last formulae to find the sum of all of the terms in a geometric series. Now, at this stage, you might be thinking that’s not possible. Surely, a series will contain infinite terms, and so we’d expect the sum to be an infinitely large number. And to an extent, this is true. For example, take the sequence two, four, eight, 16, and 32. The common ratio here is two. And so, as we progress through this sequence, the terms get larger and larger. It follows then that the sum of all of our terms would not be a number that we could evaluate.

But what about this sequence 32, 16, eight, four, two, and so on? This time, the common ratio is one-half. And so, each time the terms halve in size. This means our terms will get smaller and smaller in size. In fact, as the number of terms in our geometric series approaches ∞, the terms themselves will approach zero. And this means that the sum of these terms will actually approach a finite value. But what’s the difference here? Why can’t we evaluate the sum of all terms in our second sequence, but not the first? Well, it’s because the terms get smaller and smaller, and this only happens when the common ratio is a number less than one.

In fact, we’ll generalize a little bit further, and we say that if the absolute value of the common ratio is less than one, then the geometric series is convergent, in other words, if 𝑟 is greater than negative one and less than one. And if the series is convergent, we’re able to then find the sum to ∞. So, how can we use this fact to find the sum of all of our terms? Let’s go back to one of our 𝑆 sub 𝑛 formulae and think about what happens as 𝑛 approaches ∞. Well, since the modulus of 𝑟 is less than one — in other words, it’s a number between negative one and one — as 𝑛 gets larger, 𝑟 to the power of 𝑛 will get smaller. As 𝑛 approaches ∞, 𝑟 to the power of 𝑛 approaches zero. And so, this means the sum to 𝑛 must approach 𝑎 times one minus zero over one minus 𝑟, which is simply 𝑎 over one minus 𝑟.

Let’s formalize this. And we can say that for a convergent geometric series, the sum to ∞ is equal to 𝑎 over one minus 𝑟. It’s really, really important to realize that if the series isn’t convergent — in other words, if the modulus of the common ratio is not less than one — we’re not able to find the sum of an infinite number of terms. Now that we have a definition, we’re going to look at a few examples. In our first, we’ll decide which sequences can be summed to ∞.

Which of the following geometric sequences can be summed up to ∞?

And then, we’re given five sequences to choose from. We begin by recalling that a geometric sequence is one which has a common ratio between terms. The 𝑛th term of a geometric sequence is 𝑎𝑟 to the power of 𝑛 minus one, where 𝑎 is the first term and 𝑟 is the common ratio. Now, we can sum a geometric sequence to ∞ if it’s convergent. And this occurs when the modulus of 𝑟 or the absolute value of 𝑟, where 𝑟 is the common ratio, is less than one.

Now, we’re told that all of these sequences are geometric sequences, and so our job is to find the common ratio. Let’s begin with sequence (A). That’s eight times six to the power of 𝑛 minus five. We’re going to need to write this in this form: 𝑎 times 𝑟 to the power of 𝑛 minus one. And we can use one of our rules for exponents to do so. We know that when we multiply two exponential terms which have the same base, we simply add their exponents. And so, we can reverse this and say that “Well, six to the power of 𝑛 minus five is the same as six to the power of negative four times six to the power of 𝑛 minus one.” Six to the power of negative four, though, is one over six to the power of four. And so, we can rewrite this as eight over six to the power of four times six to the power of 𝑛 minus one.

If we compare this to the general 𝑛th term of a geometric sequence, we see that 𝑎, which is the first term in this sequence, is eight over six to the fourth power. 𝑟, however, is six. Now, it’s quite clear to us that six is not less than one. We can, therefore, say that sequence (A) is not convergent, and so we cannot sum it up to ∞.

And so, we move on to sequence (B). This time, we recall that we can find the common ratio by dividing any term by the term that precedes it. And so, in this case, we can divide three twenty-eighths by one twenty-eighth. And since the denominators of these fractions is the same, that’s three divided by one, which is three. Once again, three is not less than one. And so, sequence (B) is not convergent and cannot be summed to ∞.

Let’s have a look at sequence (C) now. Once again, we’ll find the common ratio by dividing the second term by the first. Remember, we would get the same answer if we divided the third term by the second and so on. One method we have for dividing fractions is to create a common denominator. Now, if we multiplied 263 by five, we get 1315. So, 263 is equivalent to one thousand three hundred and fifteen fifths. And so, we then divide the numerators. And we find 𝑟 is equal to negative 789 over 1315. This value for 𝑟 is between negative one and one. And so, we can say that the absolute value of the common ratio in this sequence is less than one. This means it’s convergent and it can be summed to ∞.

Let’s just double-check the other two sequences. By dividing the second term by the first in sequence (D), we can see that the absolute value of the common ratio is not less than one, and so it can’t be (D). And in fact, we get the same value for the common ratio in sequence (E). And so that confirms to us that the only geometric sequence in this list that can be summed to ∞ is (C).

In our next example, we’ll find the sum of a geometric series.

Find the sum of the geometric series 13 over two plus 13 over four plus 13 over eight, and so on.

We’re not told the number of terms to find the sum of on this geometric series. And so, we need to assume that we’re going to find the sum of all of the terms. In other words, we need to find the sum to ∞ of this series. So, we recall that for a geometric series with first term 𝑎 and common ratio 𝑟, the sum to ∞ is 𝑎 over one minus 𝑟. But this formula can only be applied if the absolute value of the common ratio is less than one. And so, the first thing that we’re going to do is just double-check that this geometric series is convergent, that the absolute value of 𝑟 is less than one.

Now, we know that, for a geometric sequence, we can find the value of the common ratio by dividing any term by the term that precedes it. Let’s divide the second term by the first. So, the common ratio is 13 over four divided by 13 over two. And one technique that we have to divide fractions is to make the denominators of those fractions the same. If we multiply the numerator and denominator of 13 over two by two, we get 26 over four. And then, once the denominators are the same, we simply divide the numerators. So, the common ratio is 13 over 26, which is equal to one-half. Now, the absolute value of one-half is just one-half, and that is indeed less than one. This means our geometric series is indeed convergent, and we’re able to find the sum to ∞.

𝑎, the first term in our series, is clearly 13 over two. And we just calculated 𝑟 to be equal to one-half. This means the sum to ∞ of our series is 13 over two divided by one minus one-half. Now, one minus one-half is one-half. So, we’re working out 13 over two divided by one-half. And since the denominators are the same, we simply divide the numerators. That tells us that the sum to ∞ is 13 divided by one, but that’s 13. Now, since the sum to ∞ is the sum of all terms of our geometric series, we’re finished. The sum of the geometric series given is 13.

We’ll now consider how to find the sum of an infinite geometric sequence given two of its terms.

Find the sum of an infinite geometric sequence given the first term is 171 and the fourth term is 171 over 64.

We know that we can find the sum of a convergent geometric sequence by using the formula sum to ∞ is 𝑎 over one minus 𝑟. And the sequence is said to be convergent if the absolute value of its ratio is less than one. Now, 𝑎 is the first term in the sequence, and we’re told that the first term is 171. But what is the common ratio? Well, we’ll use the general formula for the 𝑛th term in a geometric sequence to find this.

This is 𝑎 times 𝑟 to the power of 𝑛 minus one, where once again 𝑎 is the first term and 𝑟 is the common ratio. We’ll combine this with the fact that the fourth term in the sequence is 171 over 64. And this means that 𝑢 sub four is 171, that’s 𝑎, times 𝑟 to the power of four minus one or 𝑟 cubed. But in fact, we know the value of 𝑢 sub four. It’s 171 over 64. And so, our equation is 171 over 64 equals 171 times 𝑟 cubed. Let’s divide both sides of this equation by 171. And so, 𝑟 cubed is equal to one over 64.

We can solve for 𝑟 by taking the cube root of both sides, and we find that 𝑟 is the cube root of one over 64, which is simply equal to one-quarter. The absolute value of one-quarter is just one-quarter; it’s less than one. And so, we’ve confirmed that the geometric sequence is indeed convergent. And we now know the value of 𝑎, that’s 171, and 𝑟.

Let’s substitute everything we know into the formula. We get 171 over one minus one-quarter as being the sum to ∞, in other words, the sum of all terms in our sequence. That’s 171 over three-quarters. Now, of course, we can divide by a fraction by multiplying by the reciprocal of that fraction. That’s the same as 171 times four over three. Then, we cross-cancel by dividing 171 and three by three, meaning we’re left with 57 times four over one, which is just 57 times four. And that’s equal to 228. And we can, therefore, say that the sum of an infinite geometric sequence with a first term of 171 and a fourth term of 171 over 64 is 228.

In our final example, we’ll look at how we can use this very same process to convert a recurring decimal to a fraction.

Express 0.375 recurring as a common fraction.

Now, at first glance, it might seem like this hasn’t got anything to do with finding the sum of a geometric series. However, let’s look at the decimal 0.375 recurring and find an alternative way to write it. We know that each of the digits three seven five recur. So, it’s 0.375375375 and so on. And then, we can split this decimal up, and we can say that it’s the sum of 0.375, 0.000375, and 0.000000375. It might even help to write each of those as a fraction, so 375 over 1000, 375 over a million, and so on.

We do indeed now have a geometric series. The common ratio in this series is one one thousandth. Now, we could convince ourselves that is true by dividing any term by the term that precedes it. And since the absolute value of one one thousandth is just one one thousandth and that is less than one, we can say that the geometric series we’ve created is convergent. And since it’s convergent, we’re able to calculate the sum to ∞. In other words, we’re able to sum all of its terms. The formula we use is 𝑎 over one minus 𝑟, where 𝑎 is the first term and 𝑟 is the common ratio.

Now, we see that the first term in our series is 375 over 1000, and 𝑟 is one over 1000. And so, our sum to ∞ must be 375 over 1000 divided by one minus one over 1000. The denominator of this fraction just simplifies to 999 over 1000. And we see that we’re dividing a pair of fractions whose denominator is equal. In this case then, we can simply calculate the quotient by dividing their numerators. That’s 375 over 999. Wherever possible, we should simplify our fraction. And we can indeed divide both the numerator and the denominator of our quotient by three. When we do, we find that 375 divided by three is 125 and 999 divided by three is 333. The sum of our geometric series then, which we said was equal to 0.375 recurring, is 125 over 333. And so, we’ve written our recurring decimal as a common fraction.

Let’s recap the key points from this lesson. In this lesson, we learned that if the absolute value of the common ratio 𝑟 of a geometric sequence is less than one, then we say that that sequence is convergent. We saw that when we find the sum of all of the terms in a geometric sequence, we call that a series. And if this is the case, if the absolute value of 𝑟 is less than one for this series, we can evaluate its sum. In other words, we can find the sum of all of its terms. We call this the sum to ∞ of the series. And it’s calculated by dividing 𝑟, 𝑎 that’s the first term, by one minus 𝑟, where 𝑟, of course, is the common ratio. Finally, we saw that by writing a recurring decimal as a geometric series, we can actually use this process to convert it into a fraction.