In this video, we'll look at
sums of infinite series.

Will stop by revising what a

finite series is. A finite
series is the sum of all the

terms of a finite sequence.

An example of a finite sequence
is something like this.

Would have two K plus one
with K going from one to

10.

And that's equal to.

What starts with three? Then
it goes 5 Seven, and

so on. 21

So an example of a finite series
is just all of these things

added together, so that's 3 + 5
+ 7 plus all the way up to

21. To do this we write Sigma,
which means the sum of and then

the rule for the cake term. So
that's 2K plus one.

And then we have to write from
where case starts from where it

goes too. So that's from K

equals 1. Up cake was 10.

In this case, the sum of this
series is 120.

An infinite series is
the sum of all the terms

of an infinite sequence.

So I'll give you an example of
an infinite sequence here we

could have. One over 2 to
the KOK equals 1.

To Infinity.

And that is.

A half a quarter.

Knife.

And so on. So the series
that goes with this sequence

would be 1/2 + 1/4 plus

an eighth. And so on.

And again we can use Sigma
notation to write this.

We put down Sigma
like we did before.

Write down the rule for the
cake turn, so that's one

over 2 to the K.

And this time K starts at one
and goes on forever. So I put an

Infinity on top of the Sigma.

Now what could we mean by the

sum of this series? Without
it, the first few terms and

see what happens.

So series
was half

plus 1/4.
Put on a few more terms, this

time 116th. One over 32.

And so on.

If I add up the first 2 terms.

I'll get 3/4.

If I add up the first three

terms. I get

7/8. If I add up
the first four terms.

I get 15 over 16.

And I found out the first 5

terms. I get 31 over 32.

The sums of these first few
terms are called partial sums.

We say the first partial sum is
just the first term on its own.

The second partial sum is the
sum of the first 2 terms, so

in this case 3/4 the third
partial sum is the sum of the

first three terms. So in this
case 7/8 and so on.

If we write down the partial

sums. So that's
a half three

quarters 7/8. 15 sixteenths
and so on.

We can see that they formed
the beginning of an

infinite sequence.

The in terms of the sequence is
the NTH partial sum.

And if you look at the sequence,
you can see that it gets closer

and closer to one, so it has

limit one. So that means as
you are done, more terms of

the series.

It gets closer and closer to

one. So it makes sense to say
that this series has some one.

So the series one over 2
KOK equals 1 to Infinity.

Which is this?

Equals 1.

If we
have a

general theories
like.

AK.

OK equals 1 to Infinity, so
that's just the Series A 1 + 82

+ 83 plus a four and so on.

We say that the series has a

sum. If the NTH partial sums
converge to a limit.

So that is the sequence
given by A1 A 1

+ 82. Say 1 + 82 + 83. These
are the NTH partial sums, so we

say that this.

Series Has a sum. If this
sequence has a limit.

Here's another infinite series
that has a sum.

This time will have.

The series one over K Times
K plus one with K going from

one to Infinity.

And that's equal to 1 over 1 *
2 first term.

And then one over 2 * 3, then
one over 3 * 4.

One over 4 * 5 and so on.

Now we can split up each
individual term of this series.

So we look at one term, one over
K Times K plus one.

And I can break up the numerator
in 2K plus one minus K.

So then I can break up the

whole fraction. In 2K plus one
over K Times K plus one.

That's minus K over K Times
K plus one.

Now you can see the first bit of

this. The whole numerator
cancels with some of the

nominator so that K Plus one
cancels with this K plus one.

And on this side that K cancels
with this K.

So we get left with on this side
one over K.

Now that side we get left with
minus one over K plus one.

So now I can substitute this
back into our original series.

To rewrite it.

So the series one over K Times K
plus one from K equals 1 to

Infinity can be rewritten like

this. Well, for the first term,
cake was one.

And we've just shown that this
can be written as one over K

minus one over K plus one.

So since K is one.

For the first term, we get one
over one which is 1 - 1 over 2.

So that's the first term. We do
the same for the second term,

this time Kay is 2, so we get

one over 2. Minus one over 2 +
1 which is 3.

We carry on doing this so the
next one is 1/3 - 1/4.

And so on.

So from this we can workout the

NTH partial sum. The sum of the
first end terms is 1 - 1/2

+ 1/2 minus third.

All the way up to the NTH term,
which is one over N minus one

over N plus one.

So this. Is the
sum of the first in terms.

Now.
You can see that here a lot of

things are counseling. You get
minus 1/2 + 1/2 - 1/3. Then you

would get plus 1/3 and so on.

So overall, you're left with 1 -
1 over N plus one.

Which is equal to N plus
one over N plus one minus

one over N plus one which
is just N over N plus one.

So that was the end partial

sum. So the sequence of partial
sums goes like this.

It's the sequence and every one
plus one from N equals 1 to

Infinity. And that goes one over

2. Then 2/3

3/4

4/5 And so on.

I can see that this sequence
gets closer and closer to one.

In fact, the sequences
limit one.

So the sequence of partial sums
for that last series has limit

one. And that must mean.

That this series.

This series
has some

one.

Here's an example of a
series that doesn't have

a son.

Will have The series.

Go from one to Infinity again.

Of just one. So that's the
same as that just goes 1 +

1 + 1 and so on forever.

If we look at the sequence of
partial sums for this, it just

goes or the first one is one,
then it's one plus one which is

2. Then it's 1 + 1 + 1 which is
3, then four, and so on.

Now, this sequence certainly
doesn't get closer to any

number. In fact, it tends
to Infinity.

If this happens, we say this
series doesn't have a sum.

You might have spotted that
both the series we looked at

that had sums had terms they
got closer to 0.

In fact, this is true always. If
a series has a son that the

terms must get closer to 0.

But this doesn't work the other
way round. Here's an example.

Will have the series.

One over K from K equals 1 to

Infinity. So that's the series.

1 + 1/2.

Plus 1/3 as a quarter.

And so on.

Now, even though these terms get

close to 0. The sequence of
partial sums for this tends to

Infinity. The series, called the
harmonic series, and it doesn't

have a sum. Now we'll have
a look at two very

useful infinite series.

First of all, we'll look at this

one. It's one over K minus

1 factorial. OK cause one to

Infinity. So that's equal to one
over not factorial, which is

one, so that's one.

Then one over 1 factorial.

Which again is one.

Then one over 2 factorial which

is 2. One over 3 factorial which

is 6. And so on.

You can use your Calculator
to work at the end partial

sums of this.

I've done this already and the
end partial sums turn out to be,

well. The first one obviously is
one, then two, then 2 1/2.

And then you start needing
to use your Calculator so

the next one is.

2.6 recovering so I'll put this
down to a few decimal places.

The Next One is 2

points. 708
Then three recurring.

And the next one is.

2.7176

recovering. Will
do one more

that's. Two points.

718015
occurring.

You can see that these numbers
are getting very close to

something just below 3.

In fact, these numbers keep
getting closer to E.

So if you calculate.

The series. For large number of

terms. You can workout a very

good approximation free. So this
makes the series very very

useful. Here's
another useful

infinite series.
This time we have.

The sum of minus one to the K

plus one. All times
for over 2K minus one.

Again, K goes from one to
Infinity here.

Let's workout the four over 2K
minus one, but first.

When case one.

This denominator becomes one.

So this is 4 over 1 which is 4.

Case 2. The nominator is 3,
so this becomes for over 3.

The Next One is 4 over 5.

We get for over 7.

Forever 9 and so on.

Now we just need to workout the
minus one to the K plus one bit.

If K is odd, so that's for the
terms 135 and so on kaizad. Then

keep this one is even.

So minus 1 two and even number
is just one.

So all these odd terms.

A plus their positive.

So put pluses in front of them.

But if K is even.

Than K Plus One is odd and minus
1 two an odd number is minus

one. So all these even terms
have a minus before them.

Again, we can we use our
calculators to workout with the

NTH partial sums of the series

are. I've done this already.

On the end, partial sums turn
out to be the first one is easy,

it's 4. The Next One is 2 and
2/3, so if I write the decimal,

it's 2.6667 to a few decimal

places. Then you really
start to need your

Calculator. The next one is
3.4 than six recurring.

The Next One is 2.895. Two
that will do goes on.

The Next One is

3.33. 97 that goes on as well.

Then 2.97.
6046 and so on.

And I'll do one more at 3.28.

37

38

Now with this sequence it's a
bit hard spot which number is

converging to. You could
probably spot it's converging to

a number just above 3.

In fact, it converges to π.

So if you want to calculate π to
a large number of decimal

places, all you have to do is to
workout a large and partial sum

of this and that will give you a
good approximation for pie.