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GILBERT STRANG: OK, I
thought I would talk
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today about power series.
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These are powers of x.
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I'm going to keep going.
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All powers, all those x to the
fourth, x to the fifth,
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they'll all come in too.
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And my idea is combine them,
add them up to get
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a function of x.
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So we're doing calculus, but a
new part of it, with these
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infinite series.
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So what do I mean
by combine them?
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I mean I'll multiply those
powers by some numbers.
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Let me call those numbers a0.
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So this first guy will be
an a0, and then I'll
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add on an a1 x.
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I'm out of x.
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I'll add on some a2 x squared,
some a3 x cubed and onwards.
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So now I have a function.
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And that function, let
me call it f of x.
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So here's my starting
plan here.
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Well, we've seen this
for e to the x.
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Let me remember how e to the x
could come, the series for
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that particular function.
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So here's the plan.
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I'm going to choose those
a's so as to match--
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let me put these words down.
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I'll match at x equals 0.
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The function, its derivative,
its next derivative, its third
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derivative, and onwards.
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Each a, like a3, will be chosen
so that that right-hand
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side has the correct derivative,
third derivative,
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at x equals 0.
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So this Taylor series--
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Taylor's name is associated
with series like this--
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everything's happening
at x equals 0.
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So in the case of e
to the x, all its
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derivatives were the same.
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Still e to the x.
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And they all equal
1 at x equals 0.
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So I want that function
to give me 1 for every
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derivative.
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That doesn't mean that the
a's should all be 1.
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Why not?
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Because when I take the
derivative, for example, of
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this guy, that x cubed,
the first
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derivative, will be 3x squared.
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The next derivative 6x.
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The next derivative 6.
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That's the one I want.
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That third derivative, but it'll
be 6 so a3 will have to
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be 1/6 to give me the
correct answer 1.
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Let me write those
things down.
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So what we just did is the
derivative of x to the n-th.
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The n-th derivative.
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What's the n-th derivative
of x to the n?
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We get to use our nice formula
for derivatives.
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So the first derivative is
nx to the n minus 1.
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The derivative of that will be
n times n minus 1 x to one
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lower power.
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Keep going, do it n times,
and what have you got?
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You finally got down to the 0-th
power of x, a constant.
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But what is that constant?
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It's n times n minus 1, so that
n-th derivative will be n
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from the first, n minus
1 from the second.
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Keep multiply until you
finally get down to 1.
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And of course, that's called
because it comes up often
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enough to have its
own special name.
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That name is n factorial.
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And it's written n with
an exclamation mark.
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So that's n factorial and that's
the n-th derivative of
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x to the n.
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So for the particular function
e to the x, if I worked out
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its series, all the derivatives
I'm trying to get
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are all 1's.
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But what the powers of x gives
me these n factorials.
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The a's had better divide
by the n factorial.
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So let me recall the series for
e to the x, and then go
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onto new functions.
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That's the point
of my lecture.
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So we're getting e to the x in
a slightly different way from
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the original way, but
this is a good way.
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e to the x at x equals 0 is 1.
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At the first derivative of e to
the x is 1, so I divide by
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1 factorial.
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That's 1.
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But here I have to divide by 2
because the second derivative
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is 2 and I want those
to cancel.
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And here I divide by--
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what do I divide by?
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6 because the third
derivative is 6.
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And a typical term is I have
to divide by n factorial
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because when I take n
derivatives I get n factorial.
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The n-th derivative of
that thing we just
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worked out is n factorial.
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So I divide by n factorial and
I've got the derivative to
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come out 1.
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And that's correct
for e to the x.
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So that's the plan, matching
derivatives at x equals 0 by
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each power of x.
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And now I'm ready for
a new function.
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And a nice choice is sine x.
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So now on this board, if I can
come here, I'm going take a
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different function.
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No longer e to the x.
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My function is going
to be sine x.
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Well, I better figure out
all its derivatives.
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And they're nice, of course.
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Sine x, its derivative.
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Can I just list them all?
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These are the things that
I have to match.
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I'll plug in x equals 0.
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But let me first find
the derivatives.
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The derivative of
sine is cosine.
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The derivative of cosine
is minus the sine.
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The derivative of minus the
sine is minus the cosine.
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And then I'm back to sine again,
and repeating forever.
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That's a list of the derivatives
of sine x.
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This is my f of x here.
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This guy, first one.
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OK, now I plug in x equals
0 because I want all the
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derivatives at 0.
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The whole series is being
built focused on that
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point x equals 0.
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So at x equals 0, that's
easy to plug in.
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The sine is 0, the cosine is
1, the minus the sine is 0.
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Minus the cosine is minus 1.
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The sine is 0.
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And repeat.
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0, 1, 0, minus 1 forever.
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OK, so I know the derivatives
that I have to match.
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Now can I construct the power
series that matches that?
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OK, so that power series will
give me sine x, and
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what will it have?
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It starts with 0.
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The constant term is 0 because
the sine of 0 when x is 0-- of
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course, we want to get
the answer is 0.
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Then, the next term, the x,
its coefficient is 1.
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1x.
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No x squared's in sine x.
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No x squared's.
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But now minus.
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Do I have minus 1x cubed?
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Not quite.
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Minus x cubed, but I have to
divide by 6 because when I
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take that three derivatives,
it will produce 6.
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So I have to divide by 6,
which is 3 factorial.
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That's really the number
that's there.
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3 times 2 times 1.
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6.
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Now the fourth degree
term, the x to the
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fourth is not there.
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x to the fifth is going to
come in with a plus.
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So there's a plus
from this guy.
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This is x to the 0,
1, 2, 3, 4, 5.
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x to the fifth.
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And now what do I
divide by now?
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5 factorial.
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120.
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And then minus and so on.
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Minus an x to the seventh
over 7 factorial.
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We have created the power series
around 0, focused on 0.
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And let me remove that 1 because
just waste of space.
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x minus x cubed.
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All odd powers and
that's because
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sine x is an odd function.
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If I change from x to minus x,
everything will change sign.
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180
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What would happen if I plugged
in x equal pi?
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Suppose I took x equal pi in
this formula for sine x.
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This infinite formula,
keeps going forever.
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00:10:22,550 --> 00:10:26,250
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Well, I would get pi minus pi
cubed over 6 plus pi to the
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fifth over 120.
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It would look ridiculous.
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But you and I know that the
answer would have to come out.
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The correct sine of pi?
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0.
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I don't plan to do it,
but it has to work.
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OK, so that's the sine.
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That's the sine.
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And it's an odd series.
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Now OK, good example.
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Its twin has got to
show up here.
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The cosine.
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What's the series
for the cosine?
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00:11:02,850 --> 00:11:05,560
These are the two series
that are worth knowing.
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00:11:05,560 --> 00:11:09,780
You notice here that slope of
1, the big deal about the
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00:11:09,780 --> 00:11:15,040
slope of sine x at x equals
0, the slope is 1.
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And that does have
a slope of 1.
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OK, what about the cosine?
203
00:11:18,990 --> 00:11:21,500
Well, now I have to plug in.
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All right, the cosine is
going to start here.
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00:11:24,460 --> 00:11:27,460
Cosine minus sine
minus cosine.
206
00:11:27,460 --> 00:11:31,720
Now my f of x is going
to be cosine x.
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00:11:31,720 --> 00:11:37,620
208
00:11:37,620 --> 00:11:41,920
And I need its derivatives.
209
00:11:41,920 --> 00:11:45,120
I'm going to have three lines
again that are going to look
210
00:11:45,120 --> 00:11:47,650
just like these three lines.
211
00:11:47,650 --> 00:11:49,490
But they'll be for the cosine.
212
00:11:49,490 --> 00:11:51,400
So they start with a cosine.
213
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Its derivative is
minus the sine.
214
00:11:54,690 --> 00:11:58,200
Its derivative is minus
the cosine.
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00:11:58,200 --> 00:11:59,600
Its derivative is what?
216
00:11:59,600 --> 00:12:09,430
Plus sine and then cosine, and
forever, minus the sine.
217
00:12:09,430 --> 00:12:11,980
And let me plug in now
at x equals 0.
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00:12:11,980 --> 00:12:16,650
219
00:12:16,650 --> 00:12:18,530
This is our system.
220
00:12:18,530 --> 00:12:21,470
Find the derivatives,
plug in 0.
221
00:12:21,470 --> 00:12:24,640
So find the derivative at 0.
222
00:12:24,640 --> 00:12:28,780
Well, the function itself,
the 0-th derivative is 1.
223
00:12:28,780 --> 00:12:30,780
The first derivative is 0.
224
00:12:30,780 --> 00:12:33,280
The second derivative
is minus 1.
225
00:12:33,280 --> 00:12:35,000
The third is 0.
226
00:12:35,000 --> 00:12:40,560
The fourth derivative is
plus 1, 0, and so on.
227
00:12:40,560 --> 00:12:47,270
It's the same line as we have,
but just starting over by 1.
228
00:12:47,270 --> 00:12:50,770
Starting with the cosine.
229
00:12:50,770 --> 00:12:54,570
I know what derivatives I
want, now I just have to
230
00:12:54,570 --> 00:13:03,660
create my series for cosine x,
which matches these numbers.
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00:13:03,660 --> 00:13:05,430
One more time.
232
00:13:05,430 --> 00:13:09,750
Just match those numbers with
the coefficients that I
233
00:13:09,750 --> 00:13:16,050
originally called a0, a1, a2,
a3, but now we have numbers.
234
00:13:16,050 --> 00:13:18,400
OK, at x equals 0.
235
00:13:18,400 --> 00:13:20,580
So how does this series start?
236
00:13:20,580 --> 00:13:24,060
At x equals 0, the
cosine of 0 is 1.
237
00:13:24,060 --> 00:13:25,200
It starts with a 1.
238
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That's the constant term
sitting there.
239
00:13:27,200 --> 00:13:30,830
240
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The coefficient of x, the
linear term is 0.
241
00:13:38,140 --> 00:13:43,560
Because the cosine has
0 slope at the start.
242
00:13:43,560 --> 00:13:46,270
Then we come to something
that shows up.
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00:13:46,270 --> 00:13:46,910
Minus.
244
00:13:46,910 --> 00:13:48,750
This will be-- now what
are we in to?
245
00:13:48,750 --> 00:13:52,270
This is the constant, the
first power is gone.
246
00:13:52,270 --> 00:13:56,200
The second power minus
x squared.
247
00:13:56,200 --> 00:14:00,980
But you know if I'm looking to
match the second derivative to
248
00:14:00,980 --> 00:14:03,280
make it b minus 1.
249
00:14:03,280 --> 00:14:06,000
Right now it's minus 2.
250
00:14:06,000 --> 00:14:09,280
Differentiating would give
me a 2x and a 2.
251
00:14:09,280 --> 00:14:14,830
So I have to divide by that
2 or 2 factorial.
252
00:14:14,830 --> 00:14:16,360
Now it's good.
253
00:14:16,360 --> 00:14:20,430
Now it matches the correct
second derivative minus 1.
254
00:14:20,430 --> 00:14:22,150
Then there's no third
derivative.
255
00:14:22,150 --> 00:14:26,470
The fourth derivative
is plus 1 x to the
256
00:14:26,470 --> 00:14:32,250
fourth over 4 factorial.
257
00:14:32,250 --> 00:14:36,580
And then minus and so on, x to
the sixth over 6 factorial.
258
00:14:36,580 --> 00:14:44,720
All even powers, so this
is an even powers.
259
00:14:44,720 --> 00:14:48,830
The 0-th, second, fourth,
sixth power.
260
00:14:48,830 --> 00:14:53,700
So it's an even function.
261
00:14:53,700 --> 00:15:00,010
That means that the cosine of
minus x is exactly the same as
262
00:15:00,010 --> 00:15:01,260
the cosine of x.
263
00:15:01,260 --> 00:15:04,750
264
00:15:04,750 --> 00:15:12,380
We get a nice little insight
on these two special groups
265
00:15:12,380 --> 00:15:17,040
for which the sine is the
perfect example of an odd
266
00:15:17,040 --> 00:15:20,030
function and the cosine
is the perfect
267
00:15:20,030 --> 00:15:23,660
example of an even function.
268
00:15:23,660 --> 00:15:27,310
Well, there's so much here.
269
00:15:27,310 --> 00:15:33,630
What happens if I cut
the series off?
270
00:15:33,630 --> 00:15:38,000
I just want to look at those
first terms to see exactly
271
00:15:38,000 --> 00:15:39,260
what they represent.
272
00:15:39,260 --> 00:15:45,650
Suppose I stop here after
the linear term.
273
00:15:45,650 --> 00:15:47,540
What do I have?
274
00:15:47,540 --> 00:15:52,250
What is that x just by itself?
275
00:15:52,250 --> 00:15:55,530
It's really 0 plus x because
there was a 0 from the
276
00:15:55,530 --> 00:15:56,870
constant term.
277
00:15:56,870 --> 00:16:00,260
That is the linear
approximation.
278
00:16:00,260 --> 00:16:03,940
That gives me the equation
of the tangent line, y
279
00:16:03,940 --> 00:16:07,800
equals x, slope 1.
280
00:16:07,800 --> 00:16:11,050
More interesting, cut
this one off.
281
00:16:11,050 --> 00:16:13,360
Cut this one off here.
282
00:16:13,360 --> 00:16:16,870
That's a very important
estimate.
283
00:16:16,870 --> 00:16:19,810
It's not the exact cosine
because the exact cosine has
284
00:16:19,810 --> 00:16:22,180
got all these later guys.
285
00:16:22,180 --> 00:16:25,290
But don't forget and I should
have said this from the very
286
00:16:25,290 --> 00:16:34,270
beginning, these n factorials
grow fast. And all the series
287
00:16:34,270 --> 00:16:38,930
that we're talking about,
because those n factorials
288
00:16:38,930 --> 00:16:43,590
grow so fast and I'm dividing by
them, I can take any x and
289
00:16:43,590 --> 00:16:46,520
I get a reasonable number.
290
00:16:46,520 --> 00:16:52,280
If I take x equaled pi, that's
this sine series gave me 0.
291
00:16:52,280 --> 00:16:56,254
What do I get if I plug
in x equals pi
292
00:16:56,254 --> 00:17:01,150
in the cosine series?
293
00:17:01,150 --> 00:17:05,020
So the cosine series, if I
plugged in x equals pi and had
294
00:17:05,020 --> 00:17:09,290
patience to go pretty far, my
numbers would be getting near
295
00:17:09,290 --> 00:17:11,920
the cosine of pi.
296
00:17:11,920 --> 00:17:14,230
Which would be minus 1.
297
00:17:14,230 --> 00:17:16,640
I don't see minus
1 coming out.
298
00:17:16,640 --> 00:17:20,109
Here is 1, minus 1/2
of pi squared.
299
00:17:20,109 --> 00:17:21,980
I don't know, that's around--
300
00:17:21,980 --> 00:17:26,250
1/2 pi squared might be
around 5 or something.
301
00:17:26,250 --> 00:17:28,319
But they knock each other off.
302
00:17:28,319 --> 00:17:32,390
They get very small and we
get the answer minus 1.
303
00:17:32,390 --> 00:17:39,170
OK, so those are two important
series and now I get to tell
304
00:17:39,170 --> 00:17:41,530
you about Euler's
great formula.
305
00:17:41,530 --> 00:17:46,730
306
00:17:46,730 --> 00:17:52,560
It connects these three series
that you've seen.
307
00:17:52,560 --> 00:17:56,800
But to make that connection
I have to bring in the
308
00:17:56,800 --> 00:18:00,280
imaginary number i.
309
00:18:00,280 --> 00:18:01,770
Is that OK?
310
00:18:01,770 --> 00:18:05,070
Just imagine a number i.
311
00:18:05,070 --> 00:18:09,270
And everybody knows what you're
supposed to imagine.
312
00:18:09,270 --> 00:18:14,100
You're supposed to imagine that
i squared is minus 1.
313
00:18:14,100 --> 00:18:17,910
314
00:18:17,910 --> 00:18:22,880
And we all know there
is no real number.
315
00:18:22,880 --> 00:18:26,370
The square of a real number is
always going to be greater or
316
00:18:26,370 --> 00:18:27,590
equal to 0.
317
00:18:27,590 --> 00:18:35,400
So let's just create a symbol
i with a rule, with the
318
00:18:35,400 --> 00:18:38,510
understanding that any time we
see i squared, I'm entitled to
319
00:18:38,510 --> 00:18:41,750
write minus 1.
320
00:18:41,750 --> 00:18:41,860
OK.
321
00:18:41,860 --> 00:18:45,070
So now, what is Euler's
great formula?
322
00:18:45,070 --> 00:18:47,760
323
00:18:47,760 --> 00:18:54,590
Euler's great formula, his
brilliant insight was make x
324
00:18:54,590 --> 00:19:00,120
in this e to the x series,
make x imaginary.
325
00:19:00,120 --> 00:19:02,540
Change x to ix.
326
00:19:02,540 --> 00:19:05,410
So make it an imaginary
number.
327
00:19:05,410 --> 00:19:11,870
So can I just take Euler's, take
Taylor's series, or oh,
328
00:19:11,870 --> 00:19:14,460
maybe Euler's out of this
too, because that
329
00:19:14,460 --> 00:19:16,770
letter e is his initial.
330
00:19:16,770 --> 00:19:18,330
Probably he did.
331
00:19:18,330 --> 00:19:23,460
So I guess that's why he found
this lovely connection.
332
00:19:23,460 --> 00:19:30,680
So if I take e to the ix and
instead of x in this series I
333
00:19:30,680 --> 00:19:34,770
put in ix, just go for it.
334
00:19:34,770 --> 00:19:36,750
Let x be imaginary.
335
00:19:36,750 --> 00:19:41,370
OK, can I write out the
series 1 plus--
336
00:19:41,370 --> 00:19:44,330
instead of x I have ix.
337
00:19:44,330 --> 00:19:49,610
And then I have 1 over 2
factorial ix squared.
338
00:19:49,610 --> 00:19:54,890
And then I have 1 over
3 factorial ix cubes.
339
00:19:54,890 --> 00:19:58,810
And 1 over 4 factorial
ix to the fourth.
340
00:19:58,810 --> 00:20:00,350
That's e to the ix.
341
00:20:00,350 --> 00:20:03,880
342
00:20:03,880 --> 00:20:07,100
OK, you say, you just
changed x to ix.
343
00:20:07,100 --> 00:20:08,710
That's all I did.
344
00:20:08,710 --> 00:20:11,380
Now, here's the point.
345
00:20:11,380 --> 00:20:17,850
Now I'm going to look at this
mess and I'm going to separate
346
00:20:17,850 --> 00:20:25,140
out the part that is real from
the part that's imaginary.
347
00:20:25,140 --> 00:20:27,820
I'm going to separate it into
its real part and its
348
00:20:27,820 --> 00:20:29,220
imaginary part.
349
00:20:29,220 --> 00:20:33,760
So what is real in this thing?
350
00:20:33,760 --> 00:20:37,550
I see one is certainly
a real number.
351
00:20:37,550 --> 00:20:41,260
Do you see the other one, the
next one that's real?
352
00:20:41,260 --> 00:20:44,170
It comes from this i squared.
353
00:20:44,170 --> 00:20:47,790
That i squared I can replace
by minus 1, perfectly real.
354
00:20:47,790 --> 00:20:53,760
So it's minus from the i squared
1 over 2 factorial.
355
00:20:53,760 --> 00:20:55,850
x squared is still there.
356
00:20:55,850 --> 00:20:57,360
The i squared was minus 1.
357
00:20:57,360 --> 00:20:58,720
That's all.
358
00:20:58,720 --> 00:21:05,600
And then would come something
from the i to the fourth.
359
00:21:05,600 --> 00:21:08,540
Because what is i
to the fourth?
360
00:21:08,540 --> 00:21:12,090
It's i squared squared
minus 1 squared.
361
00:21:12,090 --> 00:21:13,620
We'd be back to plus 1.
362
00:21:13,620 --> 00:21:15,470
So plus sign.
363
00:21:15,470 --> 00:21:16,100
Good.
364
00:21:16,100 --> 00:21:24,070
Now comes the part that has an
i in it and a single i I have
365
00:21:24,070 --> 00:21:25,430
to live with.
366
00:21:25,430 --> 00:21:29,780
So that i is multiplied by x.
367
00:21:29,780 --> 00:21:32,250
Now I have i cubed.
368
00:21:32,250 --> 00:21:34,600
How do I deal with i cubed?
369
00:21:34,600 --> 00:21:40,070
i cubed is i squared
minus 1 times i.
370
00:21:40,070 --> 00:21:42,490
i squared times i is minus i.
371
00:21:42,490 --> 00:21:47,950
So I have a minus i.
372
00:21:47,950 --> 00:21:52,370
1 over 3 factorial and the
x cubed and so on.
373
00:21:52,370 --> 00:21:56,800
374
00:21:56,800 --> 00:21:59,230
Do you see what we have?
375
00:21:59,230 --> 00:22:04,440
Do you see what this real
part of e to the ix is?
376
00:22:04,440 --> 00:22:06,970
It's the cosine.
377
00:22:06,970 --> 00:22:09,080
Right there, same thing.
378
00:22:09,080 --> 00:22:17,040
So I'm getting cosine x for the
real part and then i times
379
00:22:17,040 --> 00:22:19,360
this series.
380
00:22:19,360 --> 00:22:22,740
And you can see what
that series is.
381
00:22:22,740 --> 00:22:30,030
It's the sine series, x minus
1/6 x cubed plus 1/20 of x to
382
00:22:30,030 --> 00:22:33,530
the fifth sine x.
383
00:22:33,530 --> 00:22:38,470
There is Euler's great formula
that e to the ix--
384
00:22:38,470 --> 00:22:42,020
oh, I better write it
on a fresh board.
385
00:22:42,020 --> 00:22:43,645
Maybe I'll just write
it over here.
386
00:22:43,645 --> 00:22:46,550
387
00:22:46,550 --> 00:22:53,720
I'm going to copy from this
board my Euler's great formula
388
00:22:53,720 --> 00:22:58,530
that e to the ix comes out to
have a real part cos x.
389
00:22:58,530 --> 00:23:02,240
Imaginary part gives
me the i sine x.
390
00:23:02,240 --> 00:23:03,420
And I'll write that down.
391
00:23:03,420 --> 00:23:05,610
Now let me work here.
392
00:23:05,610 --> 00:23:14,810
e to the ix is cos
x plus i sine x.
393
00:23:14,810 --> 00:23:17,090
And I want to draw a picture.
394
00:23:17,090 --> 00:23:18,340
OK, here's a picture.
395
00:23:18,340 --> 00:23:23,000
396
00:23:23,000 --> 00:23:28,110
Actually, Euler often wrote his
formula, or we often write
397
00:23:28,110 --> 00:23:33,006
his formula because we're taking
cosines and sines.
398
00:23:33,006 --> 00:23:35,900
Somehow x isn't such--
399
00:23:35,900 --> 00:23:37,150
those are angles.
400
00:23:37,150 --> 00:23:40,540
401
00:23:40,540 --> 00:23:40,840
So it's more natural
to write--
402
00:23:40,840 --> 00:23:44,160
Now that we've showing up with
sines and cosines, it's more
403
00:23:44,160 --> 00:23:48,360
natural to write a symbol
that we think of as
404
00:23:48,360 --> 00:23:50,570
an angle like theta.
405
00:23:50,570 --> 00:23:55,360
So you would more often
see it this way.
406
00:23:55,360 --> 00:24:00,750
I'm just changing letters from
x to theta as a way of
407
00:24:00,750 --> 00:24:03,000
remembering that
it's an angle.
408
00:24:03,000 --> 00:24:04,880
And now I'll draw it.
409
00:24:04,880 --> 00:24:09,040
So I have to draw that thing.
410
00:24:09,040 --> 00:24:13,560
OK, this is the real direction
and that's
411
00:24:13,560 --> 00:24:15,440
the imaginary direction.
412
00:24:15,440 --> 00:24:18,790
413
00:24:18,790 --> 00:24:20,670
I just go that's
the real part.
414
00:24:20,670 --> 00:24:23,110
I go cos theta across here.
415
00:24:23,110 --> 00:24:25,860
So let that be cos theta.
416
00:24:25,860 --> 00:24:29,590
And then I go upwards in the
imaginary up or down.
417
00:24:29,590 --> 00:24:33,970
So across is the real part,
up/down is the imaginary part.
418
00:24:33,970 --> 00:24:36,930
Say sine theta I go up.
419
00:24:36,930 --> 00:24:44,390
That height is sine theta
and that angle is theta.
420
00:24:44,390 --> 00:24:46,490
Fantastic.
421
00:24:46,490 --> 00:24:50,320
That's a picture of
Euler's formula.
422
00:24:50,320 --> 00:24:53,350
Well, that's the best
way to see it is
423
00:24:53,350 --> 00:24:55,880
that beautiful statement.
424
00:24:55,880 --> 00:24:58,040
And this is a picture
to remind us.
425
00:24:58,040 --> 00:25:04,860
426
00:25:04,860 --> 00:25:08,890
We would say that's the complex
plane because points
427
00:25:08,890 --> 00:25:12,950
have two parts, a real part
and an imaginary part.
428
00:25:12,950 --> 00:25:15,930
Nothing so complex about that.
429
00:25:15,930 --> 00:25:22,480
Now, before I stop, we've done
three important series.
430
00:25:22,480 --> 00:25:28,290
Can I mention two more, just two
more out of a long list of
431
00:25:28,290 --> 00:25:29,610
possibilities?
432
00:25:29,610 --> 00:25:33,410
One is the most important
series of all, where the
433
00:25:33,410 --> 00:25:35,720
coefficients are all 1's.
434
00:25:35,720 --> 00:25:38,980
435
00:25:38,980 --> 00:25:41,500
So the coefficients
are all 1's.
436
00:25:41,500 --> 00:25:43,720
That's called the geometric
series.
437
00:25:43,720 --> 00:25:45,110
Let me write its name here.
438
00:25:45,110 --> 00:25:51,590
439
00:25:51,590 --> 00:25:53,080
That's a Taylor series.
440
00:25:53,080 --> 00:25:55,190
That's a power series.
441
00:25:55,190 --> 00:25:59,710
And the function it comes
from happens to be 1
442
00:25:59,710 --> 00:26:02,670
over 1 minus x.
443
00:26:02,670 --> 00:26:04,220
That's the function.
444
00:26:04,220 --> 00:26:08,110
And you will see why, if you
multiply both sides by 1 minus
445
00:26:08,110 --> 00:26:10,970
x, I'll get 1 here.
446
00:26:10,970 --> 00:26:14,360
If you watch, everything will
cancel except the 1.
447
00:26:14,360 --> 00:26:16,090
So that's it.
448
00:26:16,090 --> 00:26:19,630
Now, there's a significant
difference between that series
449
00:26:19,630 --> 00:26:20,940
and e to the x.
450
00:26:20,940 --> 00:26:24,180
The biggest difference is
we're not dividing by n
451
00:26:24,180 --> 00:26:27,310
factorial anymore.
452
00:26:27,310 --> 00:26:32,790
And as a result, these terms
don't get necessarily smaller
453
00:26:32,790 --> 00:26:34,430
and smaller and smaller.
454
00:26:34,430 --> 00:26:36,410
Unless x is below 1.
455
00:26:36,410 --> 00:26:42,640
So we're OK for x below 1.
456
00:26:42,640 --> 00:26:44,530
And x could be negative.
457
00:26:44,530 --> 00:26:49,270
I can even say absolute value of
x below 1, then these terms
458
00:26:49,270 --> 00:26:49,930
gets smaller.
459
00:26:49,930 --> 00:26:52,700
But at x equals 1 we're dead.
460
00:26:52,700 --> 00:26:56,030
At x equals 1 I have 1
plus 1 plus 1 plus 1.
461
00:26:56,030 --> 00:26:57,340
All 1's.
462
00:26:57,340 --> 00:26:59,510
I'm getting infinity.
463
00:26:59,510 --> 00:27:02,240
And on the left side I'm
getting infinity also.
464
00:27:02,240 --> 00:27:05,590
At x equals 1 blows up.
465
00:27:05,590 --> 00:27:08,030
OK, one more series,
then we're done.
466
00:27:08,030 --> 00:27:10,630
One more.
467
00:27:10,630 --> 00:27:15,080
It's a neat one because it
brings in the logarithm.
468
00:27:15,080 --> 00:27:17,440
How am I going to get it?
469
00:27:17,440 --> 00:27:22,120
I'm going to start with this
series, which is the big one,
470
00:27:22,120 --> 00:27:23,840
the geometric series.
471
00:27:23,840 --> 00:27:28,170
And I'm going to take the
integral of every term.
472
00:27:28,170 --> 00:27:32,160
So if I integrate 1 I get x.
473
00:27:32,160 --> 00:27:35,460
If I integrate x I get
x squared over 2.
474
00:27:35,460 --> 00:27:39,210
If I integrate x squared
I get x cube over 3.
475
00:27:39,210 --> 00:27:41,530
x fourth over 4 and so on.
476
00:27:41,530 --> 00:27:44,800
477
00:27:44,800 --> 00:27:47,690
Not 3 factorial, just 3.
478
00:27:47,690 --> 00:27:52,880
And if I integrate this, well,
let me put the answer down and
479
00:27:52,880 --> 00:27:57,080
then we can take its derivative
and say, yep, it
480
00:27:57,080 --> 00:27:58,310
does give that.
481
00:27:58,310 --> 00:28:02,260
So the answer is minus.
482
00:28:02,260 --> 00:28:06,020
This minus sign shows up
here as a minus the
483
00:28:06,020 --> 00:28:09,010
logarithm of 1 minus x.
484
00:28:09,010 --> 00:28:12,140
485
00:28:12,140 --> 00:28:17,540
Because if I take the derivative
of that the
486
00:28:17,540 --> 00:28:21,190
logarithm always puts this
inside function down to the
487
00:28:21,190 --> 00:28:24,110
bottom, and then the derivative
of the inside
488
00:28:24,110 --> 00:28:28,350
function, the chain rule brings
out a minus 1, and the
489
00:28:28,350 --> 00:28:31,720
minus 1's go away,
and beautiful.
490
00:28:31,720 --> 00:28:37,230
So just have a look at that
series then for the logarithm.
491
00:28:37,230 --> 00:28:39,170
The logarithm of 1 minus x.
492
00:28:39,170 --> 00:28:43,300
Again, we're matching
at x equals 0.
493
00:28:43,300 --> 00:28:46,040
At x equals 0, this
function is OK.
494
00:28:46,040 --> 00:28:48,750
In fact, at x equals 0,
what is that function?
495
00:28:48,750 --> 00:28:52,670
Logarithm of 1, which is 0, and
there's no constant term.
496
00:28:52,670 --> 00:28:54,360
Good.
497
00:28:54,360 --> 00:29:00,570
OK, what comments to make about
this final example?
498
00:29:00,570 --> 00:29:04,490
This one was OK for
x smaller than 1.
499
00:29:04,490 --> 00:29:07,750
But then it died
at x equals 1.
500
00:29:07,750 --> 00:29:13,890
This one, well, it's getting
a little help dividing
501
00:29:13,890 --> 00:29:16,120
by 3 and 4 and 5.
502
00:29:16,120 --> 00:29:19,670
But that's puny help.
503
00:29:19,670 --> 00:29:24,150
That's no way compared to
dividing by 3 factorial, 4
504
00:29:24,150 --> 00:29:27,270
factorial, and so on, which
will really help.
505
00:29:27,270 --> 00:29:32,730
So actually, this series
is also only OK
506
00:29:32,730 --> 00:29:35,350
out to x equals 1.
507
00:29:35,350 --> 00:29:39,760
At x equals 1, it fails again.
508
00:29:39,760 --> 00:29:42,800
At x equals 1, what do I have?
509
00:29:42,800 --> 00:29:49,240
When x is 1, I have the log
of 0 minus infinity.
510
00:29:49,240 --> 00:29:51,850
I've got infinity
at x equals 1.
511
00:29:51,850 --> 00:29:55,020
At x equals 1, this is
1 plus 1/2 plus 1/3
512
00:29:55,020 --> 00:29:57,230
plus 1/4 plus 1/5.
513
00:29:57,230 --> 00:30:00,640
Getting smaller, but
not very fast and
514
00:30:00,640 --> 00:30:03,060
adding up to infinity.
515
00:30:03,060 --> 00:30:05,430
So there's a whole discussion.
516
00:30:05,430 --> 00:30:12,170
We could spend hours on that
famous series, 1 plus 1/2 plus
517
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1/3 plus a quarter and other
series of numbers.
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I wanted to do calculus,
derivatives, integrals, so I
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00:30:21,960 --> 00:30:26,840
took functions and series of
powers, not series of numbers
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00:30:26,840 --> 00:30:28,830
to illustrate this.
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00:30:28,830 --> 00:30:29,660
OK, good.
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00:30:29,660 --> 00:30:31,520
Thanks.
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ANNOUNCER: This has been
a production of MIT
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00:30:33,310 --> 00:30:35,710
OpenCourseWare and
Gilbert Strang.
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00:30:35,710 --> 00:30:37,980
Funding for this video was
provided by the Lord
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00:30:37,980 --> 00:30:39,190
Foundation.
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00:30:39,190 --> 00:30:42,330
To help OCW continue to provide
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00:30:42,330 --> 00:30:45,400
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00:30:45,400 --> 00:30:46,960
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00:30:46,960 --> 00:30:49,097