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

This video is about infinite
sequences and their limits.

We'll start by revising what a

simple sequences. You should
have seen that a simple sequence

is a finite list of numbers,
like this one.

It could be something
like 135.

So on up to 19

say. Another possible
example would be something

like four 916.

Perhaps stopping it somewhere

like 81? The numbers in the
sequence. I called the terms of

the sequence. So in the second
example here, we would say that

four is the first term.

And nine is the second
term.

And so on.

An infinite sequence like simple
sequence is a list of numbers.

But an infinite sequence goes on

forever. So an infinite sequence
could be something like 258.

And this time the terms just
keep on going.

Now, if you see three

dots. Followed by something that
just means I've left at some of

the terms, so that will indicate

a finite sequence. But if you

see three dots. And nothing
after them that indicates

the terms go on forever. So
that's an infinite sequence.

We say two sequences at the same
if all the terms of the same.

This means that the sequences
must contain the same numbers in

the same places.

So if I have an infinite
sequence like 1234 and so on.

This is not the same
as the sequence that goes

2143. And so on.

Because even though the sequence
has the same numbers, the

numbers aren't falling in the
same place is so these sequences

are not the same.

The first 2 sequences are
written here have nice

obvious rules for getting the
NTH term of the sequence.

So you get the first term in the
first sequence. You take 1 * 2

and takeaway one that gives you

one. To get the second term you
take 2 * 2 to get the four and

take away one. And this rule
will work for every term of the

sequence. So we say the NTH

term. Is 2 N minus one.

Similarly. The NTH term of
the second sequence here is N

plus one all squared.

The infinite sequence here
also has a rule for getting

the NTH term.

Here we take the number of the
term multiplied by three and

take off 1.

So the NTH term.

Is 3 N minus one.

But not all sequences have a
rule for getting the NTH term.

We can have a sequence that
looks really random like.

I'd say root 3
- 599.7 and so

on. Now, there's certainly no
obvious rule for getting the

NTH term for the sequence, but
it's still a sequence.

Now let's look at some
notation for sequences.

A common way to the notice
sequence is to write the NTH

term in brackets. So for the
finite sequence, the first one

here. The NTH term
is 2 and minus one.

So we write that in brackets.

And we also need to show how
many terms the sequence has.

So we say the sequence runs from
N equals 1.

And the last term in the
sequence happens to be the

10th term, so we put a tent
up here to show that the

last term is the 10th term.

Here. For the second sequence.

The rule is N plus one squared,
so we want that all in brackets.

And this time the sequence runs
from N equals 1.

Up to that's the eighth term, so
we put Nate here.

We denote the infinite sequence
in a similar way.

Again, we put the NTH term in

brackets. So that's three N
minus one, all in brackets.

And again we started the first
time, so that's an equals 1.

But to show that the
sequence goes on forever, we

put an Infinity here.

From now on will
just focus on infinite

sequences. We're often very
interested in what happens to a

sequence as N gets large. There
are three particularly

interesting things will look at.

We look at first of

all sequences. That tends

to Infinity.
Also

sequences.
Not

10s. Minus

Infinity.
And

finally,

sequences. That
tends to a real limit.

First we look at
sequences that tend to

Infinity. We say a
sequence tends to Infinity.

If however, large number I
choose, the sequence will

eventually get bigger than that
number and stay bigger than that

number. So for plot a graph to

show you what I mean. You
can see a sequence tending

to Infinity.

So what I'll do here is.

A put the values of N.

On the X axis.

And then I'll put the value of
the NTH term of the sequence on

the Y axis.

So a sequence that tends
to Infinity looks

something like this.

Now the terms here are
getting larger and larger and

larger, and I've hit the
point from going off the

page now, but.

If I could draw a line anywhere
parallel to the X axis.

Like this one.

Then for the sequence to tend to
Infinity, we need the terms

eventually to go above that and
stay above that. It doesn't

matter how large number I
choose, these terms must go and

stay above that number for the
sequence to tend to Infinity.

Here's an example of a sequence
that tends to Infinity.

Will have the sequence
that's an squared going from

N equals 1 to Infinity.

So that starts off going
1, four 916.

And so on.

So I can plot a graph of the

sequence. I won't bother putting
in the valleys friend. I'll just

plots the values of the terms of

the sequence. So.

The sequence. Will look
something like this.

Now you can see that however
large number I choose, the terms

of the sequence will definitely
go above and stay above it,

because this sequence keeps on
increasing and it increases very

fast. So this sequence
definitely tends to Infinity.

Now, even if a sequence
sometimes goes down, it can

still tend to Infinity.

Sequence that looks a bit
like this.

Starts of small goes up for
awhile, comes back down and then

goes up for awhile again.

Comes down not so far.

And carries on going up.

Now this sequence does sometimes

come down. But it always goes
up again and it would always

get above any number I choose
and it will always stay above

that as well. So this sequence
also tends to Infinity, even

though it decreases sometimes.

Now here's a sequence that
doesn't tend to Infinity, even

though it always gets bigger.

Will start off with
the first time, the

sequence being 0.

And then a lot of 100. So this
is a very big scale here.

But here. Then I'll add on half
of the hundreds or out on 50.

Aladdin half of 50 which is 25.

It'll keep adding on half the
previous Mount I did.

And this sequence.

Does this kind of thing?

And The thing is, it never ever
gets above 200.

So we found a number here that
the sequence doesn't go above

and stay above, so that must
mean the sequence doesn't tend

to Infinity. And finally, here's
an example of a sequence that

doesn't tend to Infinity, even

though. It gets really,
really big.

Will start off with the first
time being 0 again, then one,

then zero, then two, then zero,

then 3. And so on.

Now eventually the sequence will
get above any number I choose.

But it never stays above because
it always goes back to zero, and

because it doesn't stay above
any number, I choose, the

sequence doesn't tend to

Infinity. Now we look at
sequences that tends to

minus Infinity.

We say a sequence tends to
minus Infinity If however

large as negative a number,
I choose the sequence goes

below it and stays below it.

So a good example is something
like the sequence minus N cubed.

From N equals 1 to Infinity.

I can sketch a graph of this.

This starts off
at minus one.

And then falls really rapidly.

So however low an
umbrella choose.

The sequence goes below it and

stays below it. So this sequence
tends to minus Infinity.

Just cause sequence goes
below any number doesn't mean

it tends to minus Infinity.
It has to stay below it. So

a sequence like this.

Which goes minus 1 + 1 -
2 + 2 and so on.

Now, even though the terms of
the sequence go below any large

negative number I choose, they
don't stay below because we

always get a positive term

again. So this sequence doesn't
tend to minus Infinity.

In fact, the sequence doesn't
tend to any limit at all.

If the sequence tends to
minus Infinity, we write it

like this. We
write XN Arrow Minus

Infinity. As end tends
to Infinity or the limit of

XN. As I intend to

Infinity. Equals minus Infinity.

Finally, we'll look at
sequences that tend to

a real limit.

We say a sequence tends to a
real limit if there's a number,

which I'll call L.

So that's.

The sequence gets closer and
closer to L and stays very

close to it, so a sequence
tending to L might look

something like this.

Now what I mean by getting
closer and staying close to L is

however small an interval I
choose around all. So let's say

I pick this tiny interval.

The sequence must eventually get
inside that interval and stay

inside the interval. It doesn't
even matter if the sequence

doesn't actually ever hit al, so
long as it gets as close as we

like to. Ellen stays as close as
we like, then that sequence

tends to L. Here's an
example of a sequence that

has a real limit.

Will have the sequence
being one over N for N

equals 1 to Infinity.

I'll sketch a graph of this.

The first time the sequence

is one. Then it goes 1/2,
then it goes to 3rd in the

quarter and so on.

I can see this sequence gets
closer and closer to 0.

And if I pick any tiny
number, the sequence will

eventually get that close to
0 and it will stay that

close 'cause it keeps going
down. So this sequence has a

real limit and that real
limit is 0.

Allow autograph of a
sequence that tends to

real limit 3.

So
put

three
here.

And I'll show you the intervals
I can choose.

Now I can choose a sequence.

That sometimes goes away from 3.

But eventually see it

gets trapped. And this

larger interval. Then it gets
trapped in the smaller interval.

And whatever interval I drew.

It would eventually get trapped.

That close to 3.

So even though this sequence
seems to go all over the place.

It eventually gets as close
as we like to three and stays

that close, so this sequence
tends to three.

But if a sequence tends to
real limit L, we write it

like this. We say XN
tends to L.

As an tends to Infinity or
the limit of XN equals L.

A Zen tends to Infinity.

If a sequence doesn't tend
to a real limit, we

say it's divergent.

So sequences that tend to
Infinity and minus Infinity

are all divergent.

But there are some sequences
that don't tend to either plus

or minus Infinity that are still

divergent. Here's an example.

Will have the sequence going
012, one 0 - 1

- 2 - 1 zero

and then. Repeating
itself like that.

And so on.

Now, this sequence certainly
doesn't get closer and stay

closer to any real number, so it
doesn't have a real limit.

But it doesn't go off to plus or

minus Infinity either. So
this sequence is divergent,

but doesn't tend to plus or
minus Infinity.

Now this sequence keeps on
repeating itself will repeat

itself forever. The sequence
like this is called periodic.

And periodic sequences are
a good example of divergent

sequences.