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.