## www.mathcentre.ac.uk/.../Limits%20of%20sequences.mp4

• 0:01 - 0:06
sequences and their limits.
• 0:06 - 0:08
We'll start by revising what a
• 0:08 - 0:13
simple sequences. You should
have seen that a simple sequence
• 0:13 - 0:17
is a finite list of numbers,
like this one.
• 0:18 - 0:22
It could be something
like 135.
• 0:23 - 0:27
So on up to 19
• 0:27 - 0:35
say. Another possible
example would be something
• 0:35 - 0:38
like four 916.
• 0:38 - 0:41
Perhaps stopping it somewhere
• 0:41 - 0:46
like 81? The numbers in the
sequence. I called the terms of
• 0:46 - 0:51
the sequence. So in the second
example here, we would say that
• 0:51 - 0:53
four is the first term.
• 0:53 - 0:57
And nine is the second
term.
• 0:59 - 1:00
And so on.
• 1:01 - 1:07
An infinite sequence like simple
sequence is a list of numbers.
• 1:07 - 1:09
But an infinite sequence goes on
• 1:09 - 1:16
forever. So an infinite sequence
could be something like 258.
• 1:17 - 1:20
And this time the terms just
keep on going.
• 1:21 - 1:24
Now, if you see three
• 1:24 - 1:29
dots. Followed by something that
just means I've left at some of
• 1:29 - 1:31
the terms, so that will indicate
• 1:31 - 1:33
a finite sequence. But if you
• 1:33 - 1:36
see three dots. And nothing
after them that indicates
• 1:36 - 1:39
the terms go on forever. So
that's an infinite sequence.
• 1:40 - 1:45
We say two sequences at the same
if all the terms of the same.
• 1:46 - 1:50
This means that the sequences
must contain the same numbers in
• 1:50 - 1:51
the same places.
• 1:52 - 1:59
So if I have an infinite
sequence like 1234 and so on.
• 2:00 - 2:07
This is not the same
as the sequence that goes
• 2:07 - 2:09
2143. And so on.
• 2:09 - 2:13
Because even though the sequence
has the same numbers, the
• 2:13 - 2:16
numbers aren't falling in the
same place is so these sequences
• 2:16 - 2:17
are not the same.
• 2:19 - 2:21
The first 2 sequences are
written here have nice
• 2:21 - 2:25
obvious rules for getting the
NTH term of the sequence.
• 2:26 - 2:31
So you get the first term in the
first sequence. You take 1 * 2
• 2:31 - 2:33
and takeaway one that gives you
• 2:33 - 2:39
one. To get the second term you
take 2 * 2 to get the four and
• 2:39 - 2:43
take away one. And this rule
will work for every term of the
• 2:43 - 2:45
sequence. So we say the NTH
• 2:45 - 2:48
term. Is 2 N minus one.
• 2:50 - 2:56
Similarly. The NTH term of
the second sequence here is N
• 2:56 - 2:58
plus one all squared.
• 2:59 - 3:04
The infinite sequence here
also has a rule for getting
• 3:04 - 3:05
the NTH term.
• 3:06 - 3:12
Here we take the number of the
term multiplied by three and
• 3:12 - 3:14
take off 1.
• 3:15 - 3:16
So the NTH term.
• 3:17 - 3:21
Is 3 N minus one.
• 3:22 - 3:27
But not all sequences have a
rule for getting the NTH term.
• 3:28 - 3:31
We can have a sequence that
looks really random like.
• 3:31 - 3:38
I'd say root 3
- 599.7 and so
• 3:38 - 3:42
on. Now, there's certainly no
obvious rule for getting the
• 3:42 - 3:45
NTH term for the sequence, but
it's still a sequence.
• 3:47 - 3:50
Now let's look at some
notation for sequences.
• 3:51 - 3:54
A common way to the notice
sequence is to write the NTH
• 3:54 - 3:58
term in brackets. So for the
finite sequence, the first one
• 3:58 - 4:02
here. The NTH term
is 2 and minus one.
• 4:03 - 4:05
So we write that in brackets.
• 4:06 - 4:10
And we also need to show how
many terms the sequence has.
• 4:11 - 4:14
So we say the sequence runs from
N equals 1.
• 4:15 - 4:18
And the last term in the
sequence happens to be the
• 4:18 - 4:21
10th term, so we put a tent
up here to show that the
• 4:21 - 4:22
last term is the 10th term.
• 4:24 - 4:27
Here. For the second sequence.
• 4:28 - 4:34
The rule is N plus one squared,
so we want that all in brackets.
• 4:34 - 4:37
And this time the sequence runs
from N equals 1.
• 4:38 - 4:42
Up to that's the eighth term, so
we put Nate here.
• 4:45 - 4:48
We denote the infinite sequence
in a similar way.
• 4:49 - 4:50
Again, we put the NTH term in
• 4:50 - 4:56
brackets. So that's three N
minus one, all in brackets.
• 4:57 - 5:00
And again we started the first
time, so that's an equals 1.
• 5:01 - 5:04
But to show that the
sequence goes on forever, we
• 5:04 - 5:05
put an Infinity here.
• 5:07 - 5:14
From now on will
just focus on infinite
• 5:14 - 5:19
sequences. We're often very
interested in what happens to a
• 5:19 - 5:22
sequence as N gets large. There
are three particularly
• 5:22 - 5:23
interesting things will look at.
• 5:24 - 5:28
We look at first of
• 5:28 - 5:32
all sequences. That tends
• 5:32 - 5:39
to Infinity.
Also
• 5:39 - 5:46
sequences.
Not
• 5:46 - 5:50
10s. Minus
• 5:50 - 5:54
Infinity.
And
• 5:54 - 5:57
finally,
• 5:57 - 6:05
sequences. That
tends to a real limit.
• 6:08 - 6:15
First we look at
sequences that tend to
• 6:15 - 6:23
Infinity. We say a
sequence tends to Infinity.
• 6:23 - 6:26
If however, large number I
choose, the sequence will
• 6:26 - 6:30
eventually get bigger than that
number and stay bigger than that
• 6:30 - 6:32
number. So for plot a graph to
• 6:32 - 6:36
show you what I mean. You
can see a sequence tending
• 6:36 - 6:36
to Infinity.
• 6:38 - 6:40
So what I'll do here is.
• 6:42 - 6:44
A put the values of N.
• 6:44 - 6:46
On the X axis.
• 6:47 - 6:54
And then I'll put the value of
the NTH term of the sequence on
• 6:54 - 6:55
the Y axis.
• 6:56 - 6:59
So a sequence that tends
to Infinity looks
• 6:59 - 7:00
something like this.
• 7:02 - 7:10
Now the terms here are
getting larger and larger and
• 7:10 - 7:17
larger, and I've hit the
point from going off the
• 7:17 - 7:19
page now, but.
• 7:20 - 7:25
If I could draw a line anywhere
parallel to the X axis.
• 7:26 - 7:27
Like this one.
• 7:29 - 7:33
Then for the sequence to tend to
Infinity, we need the terms
• 7:33 - 7:37
eventually to go above that and
stay above that. It doesn't
• 7:37 - 7:41
matter how large number I
choose, these terms must go and
• 7:41 - 7:45
stay above that number for the
sequence to tend to Infinity.
• 7:46 - 7:51
Here's an example of a sequence
that tends to Infinity.
• 7:51 - 7:55
Will have the sequence
that's an squared going from
• 7:55 - 7:58
N equals 1 to Infinity.
• 7:59 - 8:05
So that starts off going
1, four 916.
• 8:05 - 8:08
And so on.
• 8:08 - 8:11
So I can plot a graph of the
• 8:11 - 8:17
sequence. I won't bother putting
in the valleys friend. I'll just
• 8:17 - 8:20
plots the values of the terms of
• 8:20 - 8:22
the sequence. So.
• 8:23 - 8:28
The sequence. Will look
something like this.
• 8:29 - 8:34
Now you can see that however
large number I choose, the terms
• 8:34 - 8:38
of the sequence will definitely
go above and stay above it,
• 8:38 - 8:41
because this sequence keeps on
increasing and it increases very
• 8:41 - 8:44
fast. So this sequence
definitely tends to Infinity.
• 8:46 - 8:51
Now, even if a sequence
sometimes goes down, it can
• 8:51 - 8:54
still tend to Infinity.
• 8:55 - 8:57
Sequence that looks a bit
like this.
• 8:59 - 9:04
Starts of small goes up for
awhile, comes back down and then
• 9:04 - 9:06
goes up for awhile again.
• 9:07 - 9:09
Comes down not so far.
• 9:09 - 9:10
And carries on going up.
• 9:12 - 9:15
Now this sequence does sometimes
• 9:15 - 9:20
come down. But it always goes
up again and it would always
• 9:20 - 9:24
get above any number I choose
and it will always stay above
• 9:24 - 9:27
that as well. So this sequence
also tends to Infinity, even
• 9:27 - 9:29
though it decreases sometimes.
• 9:31 - 9:36
Now here's a sequence that
doesn't tend to Infinity, even
• 9:36 - 9:39
though it always gets bigger.
• 9:40 - 9:47
Will start off with
the first time, the
• 9:47 - 9:50
sequence being 0.
• 9:51 - 9:54
And then a lot of 100. So this
is a very big scale here.
• 9:55 - 10:00
But here. Then I'll add on half
of the hundreds or out on 50.
• 10:01 - 10:04
Aladdin half of 50 which is 25.
• 10:05 - 10:08
It'll keep adding on half the
previous Mount I did.
• 10:08 - 10:10
And this sequence.
• 10:11 - 10:13
Does this kind of thing?
• 10:14 - 10:19
And The thing is, it never ever
gets above 200.
• 10:21 - 10:25
So we found a number here that
the sequence doesn't go above
• 10:25 - 10:28
and stay above, so that must
mean the sequence doesn't tend
• 10:28 - 10:35
to Infinity. And finally, here's
an example of a sequence that
• 10:35 - 10:38
doesn't tend to Infinity, even
• 10:38 - 10:41
though. It gets really,
really big.
• 10:43 - 10:48
Will start off with the first
time being 0 again, then one,
• 10:48 - 10:50
then zero, then two, then zero,
• 10:50 - 10:54
then 3. And so on.
• 10:55 - 11:00
Now eventually the sequence will
get above any number I choose.
• 11:01 - 11:06
But it never stays above because
it always goes back to zero, and
• 11:06 - 11:09
because it doesn't stay above
any number, I choose, the
• 11:09 - 11:11
sequence doesn't tend to
• 11:11 - 11:17
Infinity. Now we look at
sequences that tends to
• 11:17 - 11:18
minus Infinity.
• 11:19 - 11:23
We say a sequence tends to
minus Infinity If however
• 11:23 - 11:26
large as negative a number,
I choose the sequence goes
• 11:26 - 11:28
below it and stays below it.
• 11:29 - 11:34
So a good example is something
like the sequence minus N cubed.
• 11:34 - 11:37
From N equals 1 to Infinity.
• 11:37 - 11:39
I can sketch a graph of this.
• 11:40 - 11:46
This starts off
at minus one.
• 11:46 - 11:50
And then falls really rapidly.
• 11:51 - 11:55
So however low an
umbrella choose.
• 11:57 - 11:59
The sequence goes below it and
• 11:59 - 12:03
stays below it. So this sequence
tends to minus Infinity.
• 12:04 - 12:09
Just cause sequence goes
below any number doesn't mean
• 12:09 - 12:15
it tends to minus Infinity.
It has to stay below it. So
• 12:15 - 12:18
a sequence like this.
• 12:20 - 12:27
Which goes minus 1 + 1 -
2 + 2 and so on.
• 12:30 - 12:34
Now, even though the terms of
the sequence go below any large
• 12:34 - 12:37
negative number I choose, they
don't stay below because we
• 12:37 - 12:39
always get a positive term
• 12:39 - 12:43
again. So this sequence doesn't
tend to minus Infinity.
• 12:44 - 12:47
In fact, the sequence doesn't
tend to any limit at all.
• 12:48 - 12:55
If the sequence tends to
minus Infinity, we write it
• 12:55 - 13:02
like this. We
write XN Arrow Minus
• 13:02 - 13:09
Infinity. As end tends
to Infinity or the limit of
• 13:09 - 13:12
XN. As I intend to
• 13:12 - 13:16
Infinity. Equals minus Infinity.
• 13:17 - 13:25
Finally, we'll look at
sequences that tend to
• 13:25 - 13:28
a real limit.
• 13:29 - 13:34
We say a sequence tends to a
real limit if there's a number,
• 13:34 - 13:35
which I'll call L.
• 13:36 - 13:37
So that's.
• 13:39 - 13:44
The sequence gets closer and
closer to L and stays very
• 13:44 - 13:48
close to it, so a sequence
tending to L might look
• 13:48 - 13:49
something like this.
• 13:56 - 14:01
Now what I mean by getting
closer and staying close to L is
• 14:01 - 14:05
however small an interval I
choose around all. So let's say
• 14:05 - 14:07
I pick this tiny interval.
• 14:13 - 14:16
The sequence must eventually get
inside that interval and stay
• 14:16 - 14:20
inside the interval. It doesn't
even matter if the sequence
• 14:20 - 14:24
doesn't actually ever hit al, so
long as it gets as close as we
• 14:24 - 14:28
like to. Ellen stays as close as
we like, then that sequence
• 14:28 - 14:35
tends to L. Here's an
example of a sequence that
• 14:35 - 14:39
has a real limit.
• 14:39 - 14:44
Will have the sequence
being one over N for N
• 14:44 - 14:46
equals 1 to Infinity.
• 14:47 - 14:50
I'll sketch a graph of this.
• 14:50 - 14:54
The first time the sequence
• 14:54 - 15:00
is one. Then it goes 1/2,
then it goes to 3rd in the
• 15:00 - 15:02
quarter and so on.
• 15:02 - 15:08
I can see this sequence gets
closer and closer to 0.
• 15:09 - 15:12
And if I pick any tiny
number, the sequence will
• 15:12 - 15:16
eventually get that close to
0 and it will stay that
• 15:16 - 15:20
close 'cause it keeps going
down. So this sequence has a
• 15:20 - 15:23
real limit and that real
limit is 0.
• 15:25 - 15:31
Allow autograph of a
sequence that tends to
• 15:31 - 15:33
real limit 3.
• 15:34 - 15:39
So
put
• 15:39 - 15:44
three
here.
• 15:45 - 15:48
And I'll show you the intervals
I can choose.
• 16:01 - 16:03
Now I can choose a sequence.
• 16:04 - 16:07
That sometimes goes away from 3.
• 16:08 - 16:11
But eventually see it
• 16:11 - 16:15
gets trapped. And this
• 16:15 - 16:20
larger interval. Then it gets
trapped in the smaller interval.
• 16:22 - 16:24
And whatever interval I drew.
• 16:25 - 16:26
It would eventually get trapped.
• 16:27 - 16:29
That close to 3.
• 16:30 - 16:33
So even though this sequence
seems to go all over the place.
• 16:33 - 16:37
It eventually gets as close
as we like to three and stays
• 16:37 - 16:39
that close, so this sequence
tends to three.
• 16:41 - 16:49
But if a sequence tends to
real limit L, we write it
• 16:49 - 16:54
like this. We say XN
tends to L.
• 16:55 - 17:03
As an tends to Infinity or
the limit of XN equals L.
• 17:04 - 17:05
A Zen tends to Infinity.
• 17:07 - 17:14
If a sequence doesn't tend
to a real limit, we
• 17:14 - 17:17
say it's divergent.
• 17:18 - 17:22
So sequences that tend to
Infinity and minus Infinity
• 17:22 - 17:23
are all divergent.
• 17:25 - 17:28
But there are some sequences
that don't tend to either plus
• 17:28 - 17:29
or minus Infinity that are still
• 17:29 - 17:31
divergent. Here's an example.
• 17:32 - 17:40
Will have the sequence going
012, one 0 - 1
• 17:40 - 17:43
- 2 - 1 zero
• 17:43 - 17:48
and then. Repeating
itself like that.
• 17:49 - 17:51
And so on.
• 17:51 - 17:54
Now, this sequence certainly
doesn't get closer and stay
• 17:54 - 17:58
closer to any real number, so it
doesn't have a real limit.
• 17:58 - 18:00
But it doesn't go off to plus or
• 18:00 - 18:04
minus Infinity either. So
this sequence is divergent,
• 18:04 - 18:07
but doesn't tend to plus or
minus Infinity.
• 18:08 - 18:11
Now this sequence keeps on
repeating itself will repeat
• 18:11 - 18:15
itself forever. The sequence
like this is called periodic.
• 18:16 - 18:19
And periodic sequences are
a good example of divergent
• 18:19 - 18:20
sequences.
Title:
www.mathcentre.ac.uk/.../Limits%20of%20sequences.mp4
Video Language:
English
 mathcentre edited English subtitles for www.mathcentre.ac.uk/.../Limits%20of%20sequences.mp4