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.