Video: Infinite Series

Video Transcript

In this video, we will learn what an infinite series is, how to represent an infinite series in sigma notation, and look at different ways to represent the same series. Let’s firstly remind ourselves about the differences between sequences and series. A sequence is a list of numbers in a specific order. A sequence can be ascending or descending.

With sequences, we have a rule for each term, for example, the sequence three, six, nine, 12, 15. If we label each term the first, second, third, fourth, and fifth terms, then we can say that the rule to find the 𝑛th term is three 𝑛. A series, on the other hand, is the sum of the terms in a sequence. For example, the series of this sequence we just looked at is three add six add nine add 12 add 15. And then the value of this series is 45. Also, series can either be finite or infinite. Let’s have a look at infinite series in more detail.

An infinite series is the sum of infinite terms that follow a rule. And so the value of a series can be infinite, such as the series one add two add three and so on. If we add the first two terms, we get three. The sum of the first three terms is six, the sum of the first four terms will be 10, and so on. And so we can see that this sum is getting larger and larger as we add a bigger number each time. So we can see that the value of this series approaches ∞.

But the really interesting thing about infinite series is that we can have infinite series which actually have a finite sum. And this actually puzzled mathematicians for a long time. And Greek philosopher Zeno came up with the following paradox to explain this.

In summary, let’s say a person is walking towards a wall. They go half the distance to the wall. And then the person goes half the remaining distance. But this is actually just a quarter of the total distance. And then they go half the remaining distance again. This is just an eighth of the total distance. And this goes on and on. And actually, then this person would never reach the wall, but they would be getting closer and closer. So what this suggests is that we can describe the total distance as the sum of infinitely smaller distances. But we say that the sum of this infinite series is one. So this is an infinite series which has a finite sum.

We represent series using special notation. This is sigma. It’s a Greek capital letter, and we use it to represent summation. 𝑎 𝑛 is the formula to find the 𝑛th term. Underneath the Σ, we indicate the first value of 𝑛. We call 𝑛 the index of summation. And one would be our first value of 𝑛. But it doesn’t have to be one. You might find that 𝑛 equals zero as the starting value. But it could be anything. And at the top of the Σ, we have the last value of 𝑛. This could be a finite number, or it could be ∞, showing that we have an infinite series. So this is the notation that we use for series. Sometimes within a theorem, you might see it written as just Σ 𝑎 𝑛. And that just means that the index doesn’t matter in this context.

Here are the two series we just looked at. We can write each of these in sigma notation. If we look at this first series here, we can see that the formula to find the 𝑛th term is just 𝑛. So we can write this series in sigma notation as Σ 𝑛. But we need to indicate our first value of 𝑛 and our last value of 𝑛. Well, the three dots here indicate that this is an infinite series. So we have ∞ on the top of the Σ. And as we start here at one, then our first value of 𝑛 must be one. So now, let’s also write this series in sigma notation.

We can spot the formula to find the 𝑛th term to be one over two raised to the 𝑛th power. We can see this as one over two to the first power is the same as one over two, one over two squared is the same as one over four, one over two cubed is the same as one over eight, and so on. So then our first value of 𝑛 is one. And as this series is infinite, on the top of the Σ, we have ∞.

Note that we don’t have to choose 𝑛 as our index. We can change it to a different letter, like 𝑘, as long as we remember to change the formula to find the 𝑛th term to 𝑘 as well. And we found already that this series is equal to one. Also note that if we’re given a series in sigma notation, we can find the first few terms. The first term when 𝑛 equals one would be four times one, the second term would be four times two, the third term would be four times three, and so on. So let’s have a look at a question involving sigma notation.

Express the series one over two add one over four add one over six add one over eight and so on using sigma notation.

Let’s recall what sigma notation looks like. We use the Greek letter Σ. And next to it, we have the formula for the 𝑛th term, which I’ll call 𝑎 𝑛. Underneath the Σ, we have the first value of 𝑛. So I’ll write 𝑛 equals 𝑘. But 𝑘 could be any number. It’s just the starting value of 𝑛. Above the Σ, we have the last value of 𝑛. Again, this could be anything; it could even be ∞.

So we’re going to express this series using sigma notation. So let’s start with writing Σ. We can see that the denominators here are going up in multiples of two. So our formula for the 𝑛th term is one over two 𝑛, starting with two times one. So underneath the Σ, we write 𝑛 equals one. And as the series has a dot dot dot, this implies we have infinite terms. So above the Σ, we write ∞ as that’s technically our last value of 𝑛. So we can express this series using sigma notation as the sum from 𝑛 equals one to ∞ of one over two 𝑛.

A useful tool which comes in handy when dealing with infinite series is index shifts. Recall that we call 𝑛 the index of summation. And we’ve seen that it doesn’t have to be the letter 𝑛; it could be a different letter. And we’ve seen that the starting value 𝑘 could actually be any number. You might often see it as 𝑛 equals zero or 𝑛 equals one, but actually it could be anything. For example, the sum from 𝑛 equals zero to ∞ of two to the 𝑛th power, the first term when 𝑛 equals zero will be two raised to the power of zero, which is just one. The second term, when 𝑛 equals one, will be two raised to the first power, which is just two. And this continues in that way.

But if instead we write the sum from 𝑛 equals one to ∞ of two to the 𝑛th power, we create a different series. The first term will be two raised to the first power because 𝑛 equals one is our starting value. So this will be two. And then it will continue on. But the important thing to note is that changing the first value of 𝑛 gives us a different series. And we can write the same series in a different way using index shifts.

For example, if we think about the series the sum from 𝑘 equals one to ∞ of 𝑎𝑟 raised to the power of 𝑘 minus one. Let’s rewrite this as a series which starts at 𝑘 equals zero. We can see that the first few terms of this sum are 𝑎𝑟 raised to the zero power add 𝑎𝑟 raised to the first power add 𝑎𝑟 raised to the second power and so on. So we could spot that actually we could rewrite this series as the sum of 𝑎𝑟 to the 𝑘th power starting at 𝑘 equals zero and going to ∞. So these are exactly the same series, but just written with a different starting value.

Let’s see another example.

Express the series the sum from 𝑛 equals one to ∞ of 𝑛 add one over 𝑛 to the fourth power as a series that starts at 𝑛 equals three.

Let’s recall what this sigma notation means. 𝑛 equals one is our starting value for 𝑛. And 𝑛 is our index of summation. The value above the Σ gives us our last value of 𝑛. And next to the Σ, we have the formula to find the 𝑛th term. But we want to start this series at a different value for 𝑛. But we don’t want to actually change the value of the series itself. If we simply change 𝑛 equals one to 𝑛 equals three, we would end up losing two terms, which would then change the value of the series. So instead, we perform an index shift.

If we start by defining a variable 𝑖 equals 𝑛 add two so that when 𝑛 equals one, 𝑖 equals three, which is what we want our new starting value to be. We can also replace the value above the Σ, the last value of 𝑛. So when 𝑛 equals ∞, 𝑖 equals ∞ plus two. But this is just ∞. So when we rewrite our series using 𝑖 equals 𝑛 add two, the value above the Σ ,∞, will stay the same. So as we’re changing our starting value for the series, we will end up changing the formula for the 𝑛th term. And we do this by replacing 𝑛 using the new variable that we defined.

Note that if 𝑖 equals 𝑛 add two, then we can say that 𝑛 equals 𝑖 minus two just by rearranging for 𝑛. So now, we’re going to rewrite our series in terms of 𝑖. So we replace 𝑛 with 𝑖. From here, we can just simplify a little bit. On the numerator here, we have 𝑖 minus two add one. So this is just 𝑖 minus one. And the denominator stays the same. Now, remember that, with sigma notation, it doesn’t matter what letter we use. So let’s change the 𝑖 to an 𝑛. So this gives us the sum from 𝑛 equals three to ∞ of 𝑛 minus one over 𝑛 minus two raised to the fourth power.

If we take the series we were given in the question and find the first few terms and then simplify them, we can compare it with the series that we found for our answer. And then again, we can simplify the terms to check that our answer gives the same series as the series we were given in the question.

Another way that we can write the same series in a different way is to remove terms from the summation. Let’s say we have the series the sum from 𝑛 equals one to ∞ of 𝑎 𝑛. Then the first term is 𝑎 one and the second term is 𝑎 two and the third term is 𝑎 three and the fourth term is 𝑎 four and so on. One thing we could do is take a term out of the summation. For example, we could remove 𝑎 one from the summation and then start the series at 𝑛 equals two. So this series will be 𝑎 two add 𝑎 three add 𝑎 four and so on. So we get the same result.

We could, of course, take the first two terms out instead and then start the series at 𝑛 equals three. This can come in handy when we want to rewrite a series in order to use a particular theorem.

But for now, let’s have a look at an example.

Which of the following series is not equivalent to the sum from 𝑛 equals one to ∞ of 𝑏 𝑛? 𝑏 one add 𝑏 two add the sum from 𝑛 equals zero to ∞ of 𝑏 𝑛 add three. 𝑏 one add the sum from 𝑛 equals two to ∞ of 𝑏 𝑛. The sum from 𝑛 equals zero to ∞ of 𝑏 𝑛 add one. 𝑏 one add 𝑏 two add the sum from 𝑛 equals one to ∞ of 𝑏 𝑛 add three. Or 𝑏 one add 𝑏 two add the sum from 𝑛 equals three to ∞ of 𝑏 𝑛.

Let’s start by having a think about what this series looks like. The first term of the series, the sum from 𝑛 equals one to ∞ of 𝑏 𝑛, will be 𝑏 one because our starting value is 𝑛 equals one. And our second term will be 𝑏 two and our third term will be 𝑏 three. And it will continue in this way. So let’s have a look at each of the five series that we’ve been given. For this first series, we have 𝑏 one add 𝑏 two. And then we’re adding the series the sum from 𝑛 equals zero to ∞ of 𝑏 𝑛 add three. So the starting value of 𝑛 is zero. So the first term of the series will be 𝑏 zero add three, which is just 𝑏 three.

The next term will be when 𝑛 equals one. So this will be 𝑏 one add three. So this will be 𝑏 four. And as this goes up to ∞, this will just continue in this way. For the second series, this starts with 𝑏 one and then we have the series the sum from 𝑛 equals two to ∞ of 𝑏 𝑛. So the first value will be when 𝑛 equals two. So this will be 𝑏 two. And then the next value will be 𝑏 three and again this goes up to ∞. So this will continue in this way.

The third series is the sum from 𝑛 equals zero to ∞ of 𝑏 𝑛 add one. So the first term will be when 𝑛 equals zero. So we’ll have 𝑏 zero add one, which is just 𝑏 one. Then when 𝑛 equals one, we’ll have 𝑏 two. And this will continue as the last value of 𝑛 is ∞. If we then have a look at the fourth series here, we have 𝑏 one add 𝑏 two. And then we have the series the sum from 𝑛 equals one to ∞ of 𝑏 𝑛 add three. So this series starts when 𝑛 equals one. And if we substitute that into the formula for the 𝑛th term, we have 𝑏 one add three, which gives us 𝑏 four. And then when 𝑛 equals two, we’ll have 𝑏 two add three, which is 𝑏 five. And the last value of 𝑛 is ∞. So this will continue in this way.

So as we’ve missed out 𝑏 three, it looks like this one is not equivalent. But let’s double check the last series to be sure. This series starts with 𝑏 one add 𝑏 two and then we have the series the sum from 𝑛 equals three to ∞ of 𝑏 𝑛. So when 𝑛 equals three, we have 𝑏 three; when 𝑛 equals four, we have 𝑏 four; and so on. So we can say that the series which is not equivalent to the sum from 𝑛 equals one to ∞ of 𝑏 𝑛 is 𝑏 one add 𝑏 two add the sum from 𝑛 equals one to ∞ of 𝑏 𝑛 add three.

So we’ve seen some infinite series which sum to ∞ and some which sum to a finite number. And in fact, there’s two really important words in mathematics which we use to describe series, convergent and divergent. A convergent series is one where the value of the series is a finite number, such as the series we saw earlier which gave us the value of one. A divergent series is one where we say the value of the series is ∞. It could be the positive or negative ∞, such as the series the sum from 𝑛 equals one to ∞ of 𝑛.

Let’s summarize the main points from this lesson. A series is the sum of the terms in a sequence. But even an infinite series can give a finite value. We usually write series using sigma notation. We can rewrite a series in a different way using index shifts or removing terms from the summation. A convergent series is one where the value of the series is a finite number. And a divergent series is one where we say the value of the series is ∞.