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 ∞.
