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PROFESSOR: Welcome
back to recitation.
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In this video I want to finish
up our work with the ratio
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test, and in particular
in this video,
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I'd like to address something
that Professor Jerison talked
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about.
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When he was talking about
these Taylor series,
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he was talking about the radius
of convergence at some point.
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And he didn't go into
it very specifically,
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but he was essentially
saying, for some values of x--
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you'll have a series that
has x to the n in it--
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for some values of x it will
converge, and for some values
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it will diverge.
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So let me just remind you of
something you already know.
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So, if we consider
the series x to the n.
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Right?
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This is the geometric
series and we
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know that this is
equal to 1 over 1 minus
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x if the absolute value
of x is less than 1.
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Right?
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And so we know that
this series is finite.
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This sum converges.
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To whatever value.
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If I plug in x
here, it converges
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when absolute value
of x is less than 1.
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And it diverges when absolute
value of x is bigger than 1.
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And we're not going to address
when the absolute value of x
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equals 1.
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We won't address that case.
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But, the point I want to make
it is that for some values
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this series converges, and
for some values this series
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diverges.
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And those values can give us
the radius of convergence.
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So the radius of
convergence of this series
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is actually 1, because x goes
from 0 up to 1, and then from 0
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down to 1.
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If you think about it, radius
might be a confusing term,
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but can think about it as a
circle in one dimension less
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than maybe you usually
think about it as a circle.
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If this is the
number line, we're
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going from 0 up to 1
and down to minus 1.
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So you're going in
one direction up to 1
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and one direction
down to minus 1.
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So this is radius 1.
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r is equal to 1.
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So what I want to
do is figure out
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how I can use the
ratio test to tell me
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the radius of convergence
of other series, other power
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series, besides this one.
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OK?
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So we're really interested
in for what values of x
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do these power series converge.
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OK?
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And how we're going
to do that, is
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we're going to directly
use the ratio test.
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And one thing that I
mentioned in the other case,
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in the other ratio
test video, is
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I said you want to have
your terms be positive.
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And so just to make this easy on
ourselves, when we do the ratio
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test we're just going to
take the absolute value
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of the ratio.
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And that will be sufficient
for our purposes,
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to determine the
radius of convergence.
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I don't want to go into anything
more complicated than that.
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So we'll see, as we
do these examples,
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we're just gonna,
we're just gonna
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take the absolute
value of the ratio.
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So let's actually
start off right away
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with doing some examples.
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And each time I
do an example, I'm
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asking the following question.
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Actually, I was going to say
for what values of x, but that's
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not quite true.
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I just want to know what is
the radius of convergence
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for each power series?
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So I'm not going to
write it up every time,
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but this is the question we
want to be thinking about.
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So I'm going to write
down some power series
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and we're going to see what
is the radius of convergence
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for each of those.
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I was going to write, find the
values of x for which the power
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series converge, but
that's not quite true
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because sometimes
it will actually
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converge on one or
both of the endpoints.
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We're not going to
deal with that one.
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But sometimes it
will converge on one
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or the other of the endpoints.
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So I'm going to be
asking the question,
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find the radius of convergence.
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So let's do an example,
and I'll show you how
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it applies to the ratio test.
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Then we'll go from there
and do some other examples.
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So let's consider the
series x over 2 to the n.
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This will be an easy one.
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Now the ratio test told us that
we examine a certain limit.
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We examine the limit
as n goes to infinity.
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It said a sub n
plus 1 over a sub n.
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Well, if we think of this
whole thing as a sub n,
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then a sub n plus 1 is x
over 2 to the n plus 1.
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And then we have to divide
by x over 2 to the n.
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So let me just make
sure we understand that.
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The way we, the way we
want to think about this
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is as if the a sub n is
actually a function of x
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that also depends on n.
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OK, so a sub n, we think of it
as x over 2 raised to the n.
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So a sub n plus 1 is x over
2 raised to the n plus 1.
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a sub n is x over
2 raised to the n.
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OK, and so this looks
exactly like the ratio
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test we had before,
I told you I'm going
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to put absolute values on it.
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And so let me simplify this.
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I put the division sign
maybe kind of funny
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'cause I wanted to
have it in a row there.
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I apologize if that
made it more confusing.
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So this equals, well
I have x to the n
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plus 1 over x to the
n times 2 to the n--
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this comes to the
numerator because it's
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in the denominator of
the denominator-- over--
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and this goes to
the denominator-- 2
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to the n plus 1.
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Now what is this?
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This whole thing
converges to what?
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2 to the n over 2 to
the n plus 1 simplifies
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to 1/2, and x to the n plus 1
over x the n simplifies to x.
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So this is equal to
absolute value of x over 2.
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Now we haven't
shown how the ratio
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test is going to help us find
the radius of convergence.
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And here's where we're
going to see it working.
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If we want to know that, we
want to draw a conclusion
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about the convergence
of this series.
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We know that if this
limit is less than 1,
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that the series
definitely converges.
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So if I take my limit
when I'm all done,
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and I set it less
than 1, and I solve
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for absolute value
of x, then that
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will tell me exactly what
the radius of convergence is.
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Let me go through
that one more time.
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We knew that if we had this
series in terms of a sub n,
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and we looked at
the ratio and we
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took the limit of these
ratios, and in the limit
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the value was
strictly less than 1,
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we know the series converges.
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So now we have this thing.
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It's in terms of x,
but ultimately it's
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the same process.
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And we get to a place
where we know the limit.
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It depends on x.
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And so the limit
will be less than 1
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exactly where this
thing is less than 1.
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Right?
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So it's now in terms of x.
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This less than 1, I'm
putting in at the end.
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This is where the ratio
test is happening.
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So what does this tell us?
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Where does this converge?
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It converges when absolute value
of x over 2 is less than 1,
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which means absolute
value of x is less than 2.
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Right?
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I just multiply by 2 'cause
absolute value of 2 is 2.
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I don't have to worry
about if 2 is negative.
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So the absolute value
of x is less than 2.
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That tells me the radius of
convergence is actually 2.
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Now this shouldn't
surprise us really.
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Because this is, in fact,
a geometric series, and we
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know that the usual
geometric series
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converges when the absolute
value of x is less than 1.
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So it's not surprising,
and actually fits
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in with what we already
know, that if this thing
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on the inside, if its
absolute value is less than 1,
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that the thing will converge.
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So that that means
absolute value of x
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has to be less than 2.
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So again, hopefully this
fits in with the knowledge
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we already have.
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Hopefully it makes sense to
you, what we're trying to do.
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And so let's do another couple
of examples to make this,
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make this solidify.
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All right, what were
my other examples?
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OK, let's do this one.
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x to the n over n factorial.
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All right, so now our
a sub n, when we look
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at this geometric-- geometric?
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This is not geometric--
when we look at the series,
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our a sub n is going to be
x to the n over n factorial.
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And so when we take
our whole limit,
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the thing that's
going to happen is
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we're going to have something
in terms of x at the end.
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Our goal is to find
for what values of x
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is that thing we
have less than 1?
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Again this is where we're
using the ratio test.
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So let's look at this thing.
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We want to look at the
limit as n goes to infinity.
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Well, a sub n plus 1 is
going to be x to the n
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plus 1 over the quantity
n plus 1 factorial.
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I'm going to put the
absolute value over there.
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Now I'm going to multiply
it by 1 over a sub n.
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So this is actually a
sub n, so 1 over that
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is n factorial over x to the n.
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Now we have to be a
little careful here.
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I want to make sure everybody
understands something.
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n factorial is n times n minus 1
times n minus 2 times n minus 3
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all the way down.
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n plus 1 factorial
is n plus 1 times
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n times n minus 1 times n
minus 2 all the way down.
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So n plus 1
factorial-- I'll just
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write this here--
n plus 1 factorial
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equals n plus 1
times n factorial.
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That's very important that
you recognize that, OK?
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Because the division
of factorials
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is a little more complicated
than the division
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of straight polynomials
or something like this.
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All right?
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Because when I take this
limit, what am I going to get?
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I'm going to get x to the
n plus 1 over x to the n.
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That's going to give me one x.
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And then n factorial divided by
n plus 1 factorial, well based
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on the fact that this is equal
to n plus 1 times n factorial,
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the n factorials divide, and
I'm left with x over n plus 1.
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That's actually what
this limit equals.
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Let me go through
that one more time.
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This part is easy,
x to the n plus 1
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over x to the n gives me the x.
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And then n factorial over n
plus 1 factorial is just 1
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over n plus 1.
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And I have be careful
because I wrote equals,
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but there's still an n here.
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So let me erase that.
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I'm not gonna, I don't think--
my studio audience didn't
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say anything yet,
but I don't think
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they were going to let
me get away with that.
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This is the limit as n goes to
infinity of x over n plus 1.
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Now what is that?
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Well as n goes to
infinity, for any fixed x
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I pick-- we have to be careful
here-- for any fixed x I pick,
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as n goes to infinity, this
quantity is equal to 0.
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OK?
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If I were moving x around,
if I were moving x with n,
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this will be a problem,
but x is fixed.
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When I do this sum, I fix
my x at the beginning.
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So for any fixed x, n plus 1
is getting arbitrarily large.
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So x over n plus 1 is
getting arbitrarily small.
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So this limit is 0.
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This is strictly less than 1.
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What does this mean?
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For any x I pick, this
whole thing is less than 1.
245
00:11:08,587 --> 00:11:09,500
Right?
246
00:11:09,500 --> 00:11:12,267
Any fixed x, this ratio
is always less than 1.
247
00:11:12,267 --> 00:11:13,100
What does that mean?
248
00:11:13,100 --> 00:11:15,660
That means the radius of
convergence is infinite.
249
00:11:15,660 --> 00:11:19,860
So the radius is infinity.
250
00:11:19,860 --> 00:11:22,230
OK.
251
00:11:22,230 --> 00:11:26,070
The radius of convergence
is actually infinite.
252
00:11:26,070 --> 00:11:28,530
What series is this?
253
00:11:28,530 --> 00:11:32,150
This is actually the Taylor
series for e to the x.
254
00:11:32,150 --> 00:11:34,440
So we know that the Taylor
series for e to the x,
255
00:11:34,440 --> 00:11:36,926
that this converges for any x.
256
00:11:36,926 --> 00:11:38,220
That's a nice thing.
257
00:11:38,220 --> 00:11:40,690
So now we've used the
ratio test to tell us
258
00:11:40,690 --> 00:11:44,590
how the Taylor series behaves
for a function that we know.
259
00:11:44,590 --> 00:11:46,680
OK, we know its radius of
convergence is infinite.
260
00:11:46,680 --> 00:11:48,210
So that's pretty nice.
261
00:11:48,210 --> 00:11:51,240
All right, let's do maybe
one or two more examples,
262
00:11:51,240 --> 00:11:56,010
depending on how
much room I have.
263
00:11:56,010 --> 00:11:59,760
OK, let's try this one.
264
00:11:59,760 --> 00:12:04,430
x to the n over n
times 2 to the n.
265
00:12:04,430 --> 00:12:07,850
All right, so this is
another power series we have.
266
00:12:07,850 --> 00:12:11,310
And we want to know, if I want
to plug in some value of x
267
00:12:11,310 --> 00:12:13,400
and I want to take
the sum, will that
268
00:12:13,400 --> 00:12:15,790
converge for that
particular value of x?
269
00:12:15,790 --> 00:12:18,586
We want to know what
values of x can I plug in.
270
00:12:18,586 --> 00:12:20,210
All right so again,
we're going to look
271
00:12:20,210 --> 00:12:23,020
at the radius of-- find
the radius of convergence
272
00:12:23,020 --> 00:12:25,860
based on the ratio test.
273
00:12:25,860 --> 00:12:35,220
So the n plus first term is x
to the n plus 1 over n plus 1 2
274
00:12:35,220 --> 00:12:37,580
to the n plus 1.
275
00:12:37,580 --> 00:12:39,930
And then I have to
multiply by the a sub n's,
276
00:12:39,930 --> 00:12:44,650
the a sub n term, or multiply
by 1 over that, sorry.
277
00:12:44,650 --> 00:12:47,980
I'm dividing by a sub
n, so I get n times 2
278
00:12:47,980 --> 00:12:51,410
to the n over x to the n.
279
00:12:51,410 --> 00:12:54,551
All right, this gives me an x.
280
00:12:57,200 --> 00:12:59,510
n plus 1 over n, let me
just write out actually
281
00:12:59,510 --> 00:13:02,310
what we get here, 2 to the
n over 2 to the n plus 1
282
00:13:02,310 --> 00:13:04,340
gives me an over 2.
283
00:13:04,340 --> 00:13:08,710
And then n over n plus 1.
284
00:13:08,710 --> 00:13:10,240
This is positive,
so I don't have
285
00:13:10,240 --> 00:13:12,700
to worry about anything there.
286
00:13:12,700 --> 00:13:16,520
The limit as n goes to
infinity absolute value of x
287
00:13:16,520 --> 00:13:18,680
over 2 times n over n plus
1, what's that equal to?
288
00:13:18,680 --> 00:13:20,920
Well this, as n goes to
infinity, is equal to 1,
289
00:13:20,920 --> 00:13:23,760
so it's absolute
value of x over 2.
290
00:13:23,760 --> 00:13:25,510
Let me again remind
you what we are doing.
291
00:13:25,510 --> 00:13:28,610
We're saying I want to know
the radius of convergence
292
00:13:28,610 --> 00:13:29,530
for this series.
293
00:13:29,530 --> 00:13:31,690
So I want to know
what radius, so what
294
00:13:31,690 --> 00:13:35,270
values of x can I put in to
make this series converge?
295
00:13:35,270 --> 00:13:37,830
I might miss the
endpoints, but beyond that,
296
00:13:37,830 --> 00:13:39,980
what values of x make
this series converge?
297
00:13:39,980 --> 00:13:41,950
So I'm looking at
the ratio test.
298
00:13:41,950 --> 00:13:44,750
I took the ratio test, I got
it all the way to a place
299
00:13:44,750 --> 00:13:46,890
where I have something
in terms of x.
300
00:13:46,890 --> 00:13:51,580
As long as that thing is
less than 1, I'm golden.
301
00:13:51,580 --> 00:13:54,060
As long as that thing is less
than 1, the series converges.
302
00:13:54,060 --> 00:13:55,643
So again, I actually
get another thing
303
00:13:55,643 --> 00:13:58,102
where the absolute value
of x is less than 2.
304
00:13:58,102 --> 00:13:58,839
All right?
305
00:13:58,839 --> 00:14:01,130
I probably should have picked
a different number there.
306
00:14:01,130 --> 00:14:04,390
What do you think would
happen if this was a 7?
307
00:14:04,390 --> 00:14:05,980
Well everything
would've been the same
308
00:14:05,980 --> 00:14:08,740
except this would have been a
7, and the radius of convergence
309
00:14:08,740 --> 00:14:10,130
would have been 7.
310
00:14:10,130 --> 00:14:12,193
So I should have picked
a different number there,
311
00:14:12,193 --> 00:14:14,026
so we had a different
radius of convergence,
312
00:14:14,026 --> 00:14:16,780
but you can see how that works.
313
00:14:16,780 --> 00:14:18,230
And then, I'm gonna do just--
314
00:14:18,230 --> 00:14:20,262
I have room, I'm going
to do one more example.
315
00:14:20,262 --> 00:14:21,970
'Cause this one's a
good one to do, also.
316
00:14:27,550 --> 00:14:31,630
OK, I'm actually going to do
another series that we know,
317
00:14:31,630 --> 00:14:35,450
x to the 2n over 2n factorial.
318
00:14:35,450 --> 00:14:37,550
I'm doing this one for
a particular reason,
319
00:14:37,550 --> 00:14:40,900
to help us deal with
when some things, when
320
00:14:40,900 --> 00:14:47,390
the exponents and the factorials
get a little more complicated.
321
00:14:47,390 --> 00:14:50,600
I want to also point
out what series is this.
322
00:14:50,600 --> 00:14:52,685
You should know
what series this is.
323
00:14:52,685 --> 00:14:54,310
And I know, without
thinking, that it's
324
00:14:54,310 --> 00:14:56,484
either sine or cosine.
325
00:14:56,484 --> 00:14:57,900
I guess I should
say it's starting
326
00:14:57,900 --> 00:15:00,002
at n equals 0 to infinity.
327
00:15:00,002 --> 00:15:02,460
And then I might get nervous
and say well, which one is it?
328
00:15:02,460 --> 00:15:04,390
Well what's the first term?
329
00:15:04,390 --> 00:15:07,460
x to the 0 is 1 over
0 factorial is 1.
330
00:15:07,460 --> 00:15:10,354
So it looks like
the first term is 1,
331
00:15:10,354 --> 00:15:11,770
so that makes me
know it's cosine.
332
00:15:11,770 --> 00:15:12,950
Right?
333
00:15:12,950 --> 00:15:15,987
If the first term were
x, I'd know it was sine.
334
00:15:15,987 --> 00:15:16,820
So if I get nervous.
335
00:15:16,820 --> 00:15:19,682
So this, where it converges,
is equal to cosine x.
336
00:15:19,682 --> 00:15:20,640
Let's look at this one.
337
00:15:23,906 --> 00:15:25,405
All right, we got,
now this you have
338
00:15:25,405 --> 00:15:27,405
to be a little more careful
because it's 2 times
339
00:15:27,405 --> 00:15:32,640
the quantity n plus 1
over 2n plus 2 factorial.
340
00:15:32,640 --> 00:15:39,940
So that's n plus 1 times
2 is 2n plus 2 times 2n
341
00:15:39,940 --> 00:15:44,342
factorial over x to the 2n.
342
00:15:44,342 --> 00:15:46,550
All right this is where it
might get a little tricky.
343
00:15:46,550 --> 00:15:49,669
This is 2n plus 2 divided by 2n.
344
00:15:49,669 --> 00:15:51,210
So this is going to
be the limit as n
345
00:15:51,210 --> 00:15:56,920
goes to infinity of x squared,
the absolute value of x squared
346
00:15:56,920 --> 00:16:00,050
which is just x squared
again, times this thing.
347
00:16:00,050 --> 00:16:02,100
So let's figure out
what this thing is.
348
00:16:02,100 --> 00:16:03,560
OK?
349
00:16:03,560 --> 00:16:04,790
What is 2n factorial?
350
00:16:04,790 --> 00:16:07,180
I'm going to write out
the first couple terms.
351
00:16:07,180 --> 00:16:12,490
OK, that's going to be 2n
times 2n minus 1 times, that's
352
00:16:12,490 --> 00:16:14,276
all the way down to 1.
353
00:16:14,276 --> 00:16:15,150
And then what's this?
354
00:16:15,150 --> 00:16:23,030
This is 2n plus 2 times
2n plus 1 times 2n times,
355
00:16:23,030 --> 00:16:24,330
all the way down to 1.
356
00:16:24,330 --> 00:16:27,910
So we see, this is what I was
talking about earlier actually,
357
00:16:27,910 --> 00:16:28,630
also.
358
00:16:28,630 --> 00:16:30,820
was that I've added 2 to this.
359
00:16:30,820 --> 00:16:33,360
So it's not surprising
I get 2 more terms,
360
00:16:33,360 --> 00:16:35,330
and then I'm down to
2n factorial again.
361
00:16:35,330 --> 00:16:35,830
Right?
362
00:16:35,830 --> 00:16:37,520
So this 2n factorial.
363
00:16:37,520 --> 00:16:39,260
This is 2n factorial.
364
00:16:39,260 --> 00:16:43,110
So the 2n factorial here divides
with the 2n factorial here,
365
00:16:43,110 --> 00:16:45,060
and I'm left with these here.
366
00:16:45,060 --> 00:16:48,370
I get 1 over 2n plus
2 times 2n plus 1.
367
00:16:48,370 --> 00:16:50,860
Now what happens as
n goes to infinity?
368
00:16:50,860 --> 00:16:53,410
Obviously this ratio goes to 0.
369
00:16:53,410 --> 00:16:55,670
So I'm actually in another
case similar to the one
370
00:16:55,670 --> 00:16:57,910
I saw with e to the x.
371
00:16:57,910 --> 00:16:58,890
Is that this goes to 0.
372
00:16:58,890 --> 00:17:00,560
So the limit is
actually equal to 0,
373
00:17:00,560 --> 00:17:02,210
which is always less than 1.
374
00:17:02,210 --> 00:17:04,670
Any value of x I
pick is less than 1.
375
00:17:04,670 --> 00:17:07,195
And so this series
actually converges for any
376
00:17:07,195 --> 00:17:09,850
x that I pick.
377
00:17:09,850 --> 00:17:12,510
So this is another case where I
have the radius of convergence
378
00:17:12,510 --> 00:17:14,142
is actually infinite.
379
00:17:14,142 --> 00:17:16,600
OK, so we'll probably give you
some problems where you have
380
00:17:16,600 --> 00:17:17,840
some other things happening.
381
00:17:17,840 --> 00:17:20,382
'Cause you can actually get
the radius convergence is 0.
382
00:17:20,382 --> 00:17:22,590
You can actually get that
the only place it converges
383
00:17:22,590 --> 00:17:23,850
is at x equals 0.
384
00:17:23,850 --> 00:17:26,540
So you might actually
get the limit
385
00:17:26,540 --> 00:17:28,349
as n goes to
infinity is infinity.
386
00:17:28,349 --> 00:17:29,890
I didn't give you
an example of that,
387
00:17:29,890 --> 00:17:32,700
but that's a case where
whatever x value you put in,
388
00:17:32,700 --> 00:17:34,930
your limit is still
bigger than 1.
389
00:17:34,930 --> 00:17:38,390
Then you would always get a
diverges, except when x is 0.
390
00:17:38,390 --> 00:17:40,850
OK, so that's another thing
that you can run into.
391
00:17:40,850 --> 00:17:42,600
But, the point I want
to make is that this
392
00:17:42,600 --> 00:17:43,975
is going to let
us determine, you
393
00:17:43,975 --> 00:17:47,810
know, at least the radius over
which these x values converge.
394
00:17:47,810 --> 00:17:49,920
And it also helps us if
we know certain things
395
00:17:49,920 --> 00:17:53,080
about a function, to tell us
where this function is actually
396
00:17:53,080 --> 00:17:56,470
equal to the series
that we're dealing with.
397
00:17:56,470 --> 00:17:58,650
So I think that's
where I'll stop,
398
00:17:58,650 --> 00:18:00,514
and I hope this was informative.