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Hi.

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Welcome back to recitation.

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We've been talking about Taylor
series for a number of

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functions and rules by which you
can compute Taylor series.

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I have here an example that I
don't think we did in lecture.

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So this is the function f
of x equals secant of x.

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Now, unlike some of the other
ones you've seen, there's not

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a really simple formula
for the whole Taylor

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series of secant x.

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So what I'd like you to do is
not to find, you know, a

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formula for the general term,
but rather, just to use some

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of the tools that we've learned
to compute the first

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few terms of the Taylor series
for f of x equals secant x.

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Say, up through the x to the
fourth term, if you wanted, or

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even a little further if you
were feeling ambitious.

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So why don't you pause the
video, have a go at that, come

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back, and we can
do it together.

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So welcome back.

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I asked you to compute
the first few

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terms of a Taylor series.

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One thing you can always do in
this case, is you can go and

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you can apply the general
formula that we have for

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Taylor series, and use it to
compute the series that way.

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So in order to do that, you
just need to compute a few

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derivatives of your function.

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So remember that the general
formula for a Taylor series is

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the Taylor series for f of x
is equal to the sum from n

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equals 0 to infinity of the nth
derivative of f taken at

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zero divided by n factorial
times x to the n.

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And since we're only interested
in the first few

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terms, this is f of 0 plus f
prime of 0 x plus f double

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prime of 0 over 2x squared,
plus dot dot dot.

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I'm not going to write out
the next few terms.

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And if you want to apply this
formula to secant of x, what

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you would have to do, is you
would have to compute these

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derivatives.

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And so we could try
doing that.

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So in our case, f of x is
equal to secant of x.

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So f of 0 is secant of
0, which is just 1.

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So then we need to know f prime
of x, so that's the

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derivative of secant of x.

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So this is one you
should remember.

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This is equal to secant
x times 10x.

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And when you plug in x equal
to 0, well, the tan x is 0.

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So f prime of 0 is equal to 0.

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And then you could keep going.

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So you could compute f
double prime of x.

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So we would have to compute
the derivative of

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secant x tan x.

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And you would do that
by the product rule.

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You know, you take derivative of
secant of x, and that gives

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you secant x tan x
times 10x plus--

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and then you, OK.

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So you leave secant of x, and
you multiply it by the

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derivative of tangent, which
is secant squared x.

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And now when you put in
0 here, you get f

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double prime of 0.

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Well, OK.

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So this has a tan x in
it, so that part's

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going to give you 0.

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And here we end up with secant
of 0 times secant squared of

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0, so that's 1 times
1 times 1.

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So that's just 1.

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So f double prime of 0 is 1, and
you could keep doing this.

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Now one thing you'll notice is
that this is getting more and

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more complicated.

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I mean, we can simplify this
expression a little bit.

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We could write it as secant x
tan squared x plus secant

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cubed x, and there are, you
know, all sorts of trig

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manipulations you could do, if
you wanted to try and rewrite

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that in some simpler form.

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But fundamentally, it's more
complicated than the first

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derivative was, and that's more

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complicated in the function.

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And it will keep getting more
complicated as you compute

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more derivatives.

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So we can do this.

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So so far this shows us, by the
way, that this is equal to

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1 plus 0x plus x squared over
2 plus dot dot dot.

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And if you wanted to compute,
you know, up through the

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fourth degree term or something
like this, that's

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something that's manageable.

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But I want to suggest that there
are maybe some nicer

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ways to do it.

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So one thing to notice is that
secant of x is closely related

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to the function cosine of x.

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And you know the Taylor series
for cosine of x.

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So one thing you could think
to do, is to leverage the

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information that you have about
cosine of x in order to

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use it to get some information
about secant of x.

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So one simple way to do that,
is that you know that cosine

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of x is even.

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It's an even function.

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Cosine of minus x is equal
to cosine of x.

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So that means secant
of x is also even.

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Secant of x is an
even function.

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And you've seen that even
functions, their Taylor

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series, all the odd powers
have coefficient 0.

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So what that means is without
ever computing the third

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derivative, we can know already
that the next term in

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this Taylor series is going
to be 0x cubed over 6.

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OK?

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So that's nice.

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So the odd terms of the Taylor
series for secant of x are 0.

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OK.

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So that's one thing you
can get right away.

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So that gives you, if you like,
that gives you half of

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the terms of the
Taylor series.

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It's a little bit
of a joke, but.

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OK, so then you only need to
figure out the even terms.

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That's one way we can leverage
the relationship between

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secant and cosine.

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The other way is that we can
remember that Taylor series

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multiply just like
polynomials do.

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So if secant is 1 over cosine,
that means secant times cosine

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is equal to 1.

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OK?

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So idea, secant of x times
cosine of x is equal to 1.

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Now, that means that the Taylor
series for secant of x

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times the Taylor series
for cosine of x has to

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simplify just to 1.

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So we can write down that
product as a product of two

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infinite polynomials,
and we can start

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multiplying term by term.

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And that'll allow us to solve
for a bunch of terms, just by

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solving some simple
linear equations.

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So let me show you
what I mean.

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So we know--

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let me write it as cosine
times secant.

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So we know that cosine of x is
1 minus x squared over 2 plus

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x to the fourth over 24.

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That's 4 factorial.

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Minus x to the sixth over 720,
which is 6 factorial

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plus dot dot dot.

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And so we know that if we
multiply this by the series

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for secant of x--

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well, what is the series
for secant of x?

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Well, we've already computed a
few terms. We know that it's 1

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plus x squared over 2, we
computed that already.

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And we know that the third
degree term is 0.

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So there's some fourth degree
coefficient a4 x to the

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fourth, or 4 factorial.

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And there's some--

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well, we know, we said it's
even, so we know the fifth

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degree coefficient is 0.

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So then the sixth degree
coefficient we don't know yet.

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So this is plus a sub six times
x to the sixth over six

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factorial, plus dot dot dot.

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So we know that when we multiply
these two things

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together, it has to
give us just 1.

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All the higher order terms have
to cancel, because over

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here we have a 1.

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So what you can do, is you can
actually try multiplying these

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things out.

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So it's easy to see, for
example, that the constant

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term of this product is just
1 times 1, which is 1.

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Which is good.

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So that's a check on what
we've done so far.

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And there is no x-term, because
there are no x's.

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The x squared term here is 1
times x squared over 2 minus x

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squared over 2 times 1.

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Well, that gives us
0, so that's good.

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So this product is equal to--
well, it's 1, plus we saw 0x,

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plus 0x squared.

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How about the x cubed term?

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Well, there are no odd terms
in this product, so

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there's no x cubed.

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How about the x to
the fourth term?

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Well, OK.

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So how do we get an
x to the fourth?

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We could have an x to the fourth
here times a constant.

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So that's x to the
fourth over 24.

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Or, we could have an x squared
times an x squared.

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So in this case, that gives us
minus x squared over 2 times x

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squared over 2, which is minus
x to the fourth over 4.

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Or we could have a constant
times an x to the fourth.

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So this is plus a4 x to
the fourth over 24.

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And then we'll have a sixth
degree term, and so on.

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Notice that there's never any
involvement from the higher

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order terms in the fourth
power here.

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Right?

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If you ever took an x to the
sixth here, well then

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everything you multiply it
by has at least an x

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to the sixth power.

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So we don't have to worry about
that showing up in the x

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to the fourth term.

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Well, OK.

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We know this is actually equal
to 1, so we know that this

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thing has to be 0.

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This is x to the fourth term.

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It has to be 0.

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So that means that 1/24
minus 1/4 for plus a4

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over 24 equals 0.

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So OK.

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So now you can multiply
everything through by 24, and

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rearrange to figure out
that a4 is equal to 5.

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So I've done that correctly.

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And then if you wanted, it would
be fairly easy to go

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back up and then you look at
the x to the sixth term.

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And from there, you could figure
out that a6 was equal

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to, say, 61 or something
like that.

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I think 61.

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And you could keep doing this.

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So this is one way to compute
more terms of the series for

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secant of x.

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Another thing you could do--
which I'll just mention very

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briefly, I'm not going to show
you how to do it-- is that you

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can do long division
on power series.

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So it actually works out very,
it works out just like this.

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It's mathematically
equivalent.

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The way you actually do
it looks different.

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It looks like long division.

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When you do long division with
polynomials, you start with

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the highest order term.

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Of course, power series don't
have highest order terms. What

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you actually do with a power
series, is you start with the

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lowest order term.

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So to divide this into 1, you'd
say, oh, you mean the

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factor of 1, and then you'll
have a, you know, you subtract

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off 1 times this, and that gives
you an x squared plus 2.

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And so, OK, so you say, I
need a plus an x squared

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plus 2, and so on.

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So that was too brief for you to
understand it properly, but

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you can go and look up somewhere
the method of long

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division on power series.

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So just to recap, we talked
about three methods for

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computing the coefficients
of a power series.

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There's the formula that you
were given, which works, and

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which you could use.

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In this case, it's a
little complicated.

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Then there's the method of using
a relationship between

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your power series and other
known power series.

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In this case, we can use the
relationship that we know our

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power series satisfies
a certain product.

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We know that our power series
times cosine of x equals to 1,

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and we can use that to solve for
some of the coefficients.

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Or you can also, similarly, when
you have that situation,

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you can also use long division
to compute the coefficients.

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I'll end there.

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