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This video is about infinite
sequences and their limits.

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We'll start by revising what a

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simple sequences. You should
have seen that a simple sequence

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is a finite list of numbers,
like this one.

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It could be something
like 135.

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So on up to 19

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say. Another possible
example would be something

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like four 916.

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Perhaps stopping it somewhere

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

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

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four is the first term.

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And nine is the second
term.

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And so on.

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

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But an infinite sequence goes on

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

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And this time the terms just
keep on going.

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Now, if you see three

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

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the terms, so that will indicate

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a finite sequence. But if you

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see three dots. And nothing
after them that indicates

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

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

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This means that the sequences
must contain the same numbers in

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the same places.

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

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This is not the same
as the sequence that goes

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2143. And so on.

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Because even though the sequence
has the same numbers, the

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

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are not the same.

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The first 2 sequences are
written here have nice

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obvious rules for getting the
NTH term of the sequence.

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

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and takeaway one that gives you

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

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

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sequence. So we say the NTH

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term. Is 2 N minus one.

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

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plus one all squared.

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The infinite sequence here
also has a rule for getting

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the NTH term.

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

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take off 1.

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So the NTH term.

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Is 3 N minus one.

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

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We can have a sequence that
looks really random like.

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I'd say root 3
- 599.7 and so

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

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

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Now let's look at some
notation for sequences.

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

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

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here. The NTH term
is 2 and minus one.

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So we write that in brackets.

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

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So we say the sequence runs from
N equals 1.

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And the last term in the
sequence happens to be the

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

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last term is the 10th term.

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Here. For the second sequence.

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

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And this time the sequence runs
from N equals 1.

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

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We denote the infinite sequence
in a similar way.

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Again, we put the NTH term in

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

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

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But to show that the
sequence goes on forever, we

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put an Infinity here.

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From now on will
just focus on infinite

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

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sequence as N gets large. There
are three particularly

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interesting things will look at.

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We look at first of

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all sequences. That tends

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to Infinity.
Also

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sequences.
Not

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10s. Minus

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Infinity.
And

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finally,

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sequences. That
tends to a real limit.

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First we look at
sequences that tend to

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Infinity. We say a
sequence tends to Infinity.

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If however, large number I
choose, the sequence will

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eventually get bigger than that
number and stay bigger than that

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number. So for plot a graph to

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

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to Infinity.

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So what I'll do here is.

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A put the values of N.

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On the X axis.

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

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the Y axis.

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So a sequence that tends
to Infinity looks

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something like this.

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Now the terms here are
getting larger and larger and

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

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page now, but.

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

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Like this one.

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

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

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

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

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

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Will have the sequence
that's an squared going from

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N equals 1 to Infinity.

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So that starts off going
1, four 916.

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And so on.

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So I can plot a graph of the

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

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plots the values of the terms of

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the sequence. So.

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The sequence. Will look
something like this.

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

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

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because this sequence keeps on
increasing and it increases very

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fast. So this sequence
definitely tends to Infinity.

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

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still tend to Infinity.

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Sequence that looks a bit
like this.

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

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goes up for awhile again.

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Comes down not so far.

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And carries on going up.

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Now this sequence does sometimes

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

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

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

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though it decreases sometimes.

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

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though it always gets bigger.

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Will start off with
the first time, the

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sequence being 0.

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

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

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Aladdin half of 50 which is 25.

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

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And this sequence.

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Does this kind of thing?

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

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

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

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

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doesn't tend to Infinity, even

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though. It gets really,
really big.

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

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then zero, then two, then zero,

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then 3. And so on.

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

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

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

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sequence doesn't tend to

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Infinity. Now we look at
sequences that tends to

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minus Infinity.

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We say a sequence tends to
minus Infinity If however

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large as negative a number,
I choose the sequence goes

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below it and stays below it.

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

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From N equals 1 to Infinity.

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I can sketch a graph of this.

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This starts off
at minus one.

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And then falls really rapidly.

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So however low an
umbrella choose.

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The sequence goes below it and

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

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Just cause sequence goes
below any number doesn't mean

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

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a sequence like this.

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

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

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

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always get a positive term

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

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

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If the sequence tends to
minus Infinity, we write it

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like this. We
write XN Arrow Minus

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Infinity. As end tends
to Infinity or the limit of

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XN. As I intend to

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Infinity. Equals minus Infinity.

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Finally, we'll look at
sequences that tend to

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a real limit.

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

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which I'll call L.

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So that's.

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The sequence gets closer and
closer to L and stays very

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

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something like this.

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

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

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I pick this tiny interval.

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The sequence must eventually get
inside that interval and stay

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

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

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

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

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has a real limit.

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Will have the sequence
being one over N for N

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equals 1 to Infinity.

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I'll sketch a graph of this.

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The first time the sequence

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

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quarter and so on.

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00:15:02,370 --> 00:15:08,079
I can see this sequence gets
closer and closer to 0.

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And if I pick any tiny
number, the sequence will

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eventually get that close to
0 and it will stay that

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

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real limit and that real
limit is 0.

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Allow autograph of a
sequence that tends to

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real limit 3.

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So
put

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three
here.

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

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Now I can choose a sequence.

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That sometimes goes away from 3.

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00:16:07,960 --> 00:16:11,480
But eventually see it

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gets trapped. And this

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

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And whatever interval I drew.

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It would eventually get trapped.

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That close to 3.

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

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

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that close, so this sequence
tends to three.

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

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like this. We say XN
tends to L.

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

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A Zen tends to Infinity.

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00:17:06,920 --> 00:17:14,380
If a sequence doesn't tend
to a real limit, we

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say it's divergent.

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00:17:18,020 --> 00:17:22,034
So sequences that tend to
Infinity and minus Infinity

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are all divergent.

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

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or minus Infinity that are still

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divergent. Here's an example.

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Will have the sequence going
012, one 0 - 1

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- 2 - 1 zero

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and then. Repeating
itself like that.

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00:17:49,090 --> 00:17:50,710
And so on.

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00:17:51,490 --> 00:17:54,109
Now, this sequence certainly
doesn't get closer and stay

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00:17:54,109 --> 00:17:57,601
closer to any real number, so it
doesn't have a real limit.

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00:17:58,240 --> 00:18:00,240
But it doesn't go off to plus or

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00:18:00,240 --> 00:18:03,730
minus Infinity either. So
this sequence is divergent,

249
00:18:03,730 --> 00:18:06,578
but doesn't tend to plus or
minus Infinity.

250
00:18:07,680 --> 00:18:11,262
Now this sequence keeps on
repeating itself will repeat

251
00:18:11,262 --> 00:18:15,259
itself forever. The sequence
like this is called periodic.

252
00:18:15,790 --> 00:18:19,435
And periodic sequences are
a good example of divergent

253
00:18:19,435 --> 00:18:19,840
sequences.