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.