Question Video: The Common Ratio of a Geometric Sequence Mathematics • Second Year of Secondary School

Video Transcript

What condition must be true of the common ratio, 𝑟, in a geometric sequence if the sum of infinitely many terms can be found?

In this question, we are asked to find the condition on the common ratio 𝑟 that guarantees that the limit of the infinite geometric series will exist.

There are many ways to answer this question. The easiest is to recall that the infinite sum of a geometric sequence with initial term 𝑎 and constant ratio 𝑟 whose absolute value is less than one is 𝑎 over one minus 𝑟. We can then recall that a geometric sequence with common ratio whose size is greater than or equal to one does not necessarily converge. So we can say that the absolute value of 𝑟 must be less than one. It is useful to be able to recall properties and theorems in this way. However, it is also important to understand why these results hold true. So, let’s analyze this question in more detail.

First, we can start by recalling that a geometric sequence is one in which we find the next term in the sequence by multiplying by a constant value of 𝑟. In other words, the ratio of successive terms in the sequence remains constant. Therefore, if the initial term in the geometric sequence is 𝑎 and the common ratio is 𝑟, we obtain a sequence 𝑎, 𝑎𝑟, 𝑎𝑟 squared, 𝑎𝑟 cubed, and so on.

Before we consider the sum of these terms, let’s start by considering the trivial cases of these types of sequences. First, if 𝑎 is zero, then every term is zero regardless of the value of 𝑟. This will not yield any useful information about the conditions of 𝑟 that make the infinite sum exist, so we can ignore this case. Similarly, if 𝑟 is zero, then every term after the first is zero. So the infinite sum is equal to the first term 𝑎. Thus, we know that any geometric sequence with 𝑟 equals zero has an infinite sum that converges.

We are now ready to consider the general case of a sum of the terms of a geometric sequence. Let’s call the sum of the first 𝑛 terms 𝑆 sub 𝑛, and we can write the sum out as shown. We can now find an expression for 𝑟 times 𝑆 sub 𝑛 by multiplying every term by 𝑟. We obtain the following expression. We can use this to find an expression for the difference between 𝑆 sub 𝑛 and 𝑟 times 𝑆 sub 𝑛. If we subtract the second equation from the first equation, we can note that all of the terms except 𝑎 and 𝑎𝑟 to the 𝑛th power cancel. This leaves us with the equation 𝑆 sub 𝑛 minus 𝑟𝑆 sub 𝑛 equals 𝑎 minus 𝑎𝑟 to the 𝑛th power.

We can rewrite this equation by taking out the shared factor of 𝑆 sub 𝑛 on the left-hand side of the equation and the shared factor of 𝑎 on the right-hand side of the equation to get the following. We can then divide the equation through by one minus 𝑟 to obtain that 𝑆 sub 𝑛 equals 𝑎 times one minus 𝑟 to the 𝑛th power over one minus 𝑟. We can note that this expression for the sum of the first 𝑛 terms is not valid when 𝑟 equals one, since we cannot divide by zero. Thus, we will have to consider this case separately.

We want to consider what happens when we take the sum of an infinite number of terms. So, we want the value of 𝑛 to approach ∞. We can see that there is only one term in this expression that changes as the value of 𝑛 changes. We know that if the size of 𝑟 is less than one, then 𝑟 to the 𝑛th power will get smaller in size as 𝑛 increases. Similarly, if the size of 𝑟 is greater than one, then the size of 𝑟 to the 𝑛th power is unbounded as 𝑛 increases. This shows two of the possible cases. When the size of 𝑟 is less than one, we can remove the term 𝑟 to the 𝑛th power in our numerator to obtain our infinite sum result. However, if the size of 𝑟 is greater than one, then the size of the terms gets larger and larger, so the sum does not converge.

There are two cases we have not yet considered. First, when 𝑟 equals one, every term in the geometric sequence is equal to 𝑎. If we then add the first 𝑛 terms of the sequence together, then we get 𝑛 times 𝑎. This will only converge as 𝑛 gets larger if 𝑎 is zero. Thus, it does not converge in general. Second, we need to consider what happens when 𝑟 equals negative one. We see that the terms of the sequence are the same size but switching in signs.

To see why the sum of infinitely many terms of this sequence does not converge in general, we can consider each sum of 𝑛 terms. This gives us a sequence called the sequence of partial sums. In this case, the sequence is 𝑎, zero, 𝑎, zero, and this sequence continues indefinitely. This only converges if 𝑎 is zero. So we can say that the sum of an infinite number of terms of a geometric sequence is guaranteed to converge if the absolute value of 𝑟 is less than one.