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PROFESSOR: So,
Professor Jerison is
00:00:26.290 --> 00:00:29.840
relaxing in sunny
London, Ontario today
00:00:29.840 --> 00:00:31.969
and sent me in as
his substitute again.
00:00:31.969 --> 00:00:33.760
I'm glad to the here
and see you all again.
00:00:38.364 --> 00:00:40.280
So our agenda today: he
said that he'd already
00:00:40.280 --> 00:00:45.200
talked about power series
and Taylor's formula,
00:00:45.200 --> 00:00:50.850
I guess on last week
right, on Friday?
00:00:50.850 --> 00:00:53.265
So I'm going to go a
little further with that
00:00:53.265 --> 00:00:57.540
and show you some examples,
show you some applications,
00:00:57.540 --> 00:00:59.900
and then I have this
course evaluation survey
00:00:59.900 --> 00:01:03.805
that I'll hand out in the last
10 minutes or so of the class.
00:01:06.890 --> 00:01:09.630
I also have this handout
that he made that says
00:01:09.630 --> 00:01:12.520
18.01 end of term 2007.
00:01:12.520 --> 00:01:15.690
If you didn't pick this up
coming in, grab it going out.
00:01:15.690 --> 00:01:18.850
People tend not to pick it
up when they walk in, I see.
00:01:18.850 --> 00:01:21.890
So grab this when
you're going out.
00:01:21.890 --> 00:01:23.390
There's some things
missing from it.
00:01:23.390 --> 00:01:27.210
He has not decided
when his office hours
00:01:27.210 --> 00:01:28.520
will be at the end of term.
00:01:28.520 --> 00:01:31.280
He will have them, just
hasn't decided when.
00:01:31.280 --> 00:01:34.575
So, check the website
for that information.
00:01:38.000 --> 00:01:42.640
And we're looking forward to
the final exam, which is uh --
00:01:42.640 --> 00:01:43.140
aren't we?
00:01:47.030 --> 00:01:49.590
Any questions about
this technical stuff?
00:01:52.900 --> 00:01:56.860
All right, let's talk about
power series for a little bit.
00:01:56.860 --> 00:02:00.000
So I thought I should
review for you what
00:02:00.000 --> 00:02:01.980
the story with power series is.
00:02:21.595 --> 00:02:23.220
OK, could I have your
attention please?
00:02:26.920 --> 00:02:31.640
So, power series is a way of
writing a function as a sum
00:02:31.640 --> 00:02:33.560
of integral powers of x.
00:02:33.560 --> 00:02:38.090
These a_0, a_1, and
so on, are numbers.
00:02:38.090 --> 00:02:43.060
An example of a power
series is a polynomial.
00:02:48.050 --> 00:02:51.250
Not to be forgotten,
one type of power series
00:02:51.250 --> 00:02:57.560
is one which goes on for
a finite number of terms
00:02:57.560 --> 00:03:02.090
and then ends, so that all of
the other, all the higher a_i's
00:03:02.090 --> 00:03:03.650
are all 0.
00:03:03.650 --> 00:03:06.150
This is a perfectly good
example of a power series;
00:03:06.150 --> 00:03:08.862
it's a very special
kind of power series.
00:03:08.862 --> 00:03:10.570
And part of what I
want to tell you today
00:03:10.570 --> 00:03:14.290
is that power series
behave, almost exactly like,
00:03:14.290 --> 00:03:14.970
polynomials.
00:03:14.970 --> 00:03:16.490
There's just one
thing that you have
00:03:16.490 --> 00:03:20.800
to be careful about when you're
using power series that isn't
00:03:20.800 --> 00:03:22.810
a concern for polynomials,
and I'll show you
00:03:22.810 --> 00:03:24.540
what that is in a minute.
00:03:24.540 --> 00:03:29.320
So, you should think of them
as generalized polynomials.
00:03:29.320 --> 00:03:32.140
The one thing that you
have to be careful about
00:03:32.140 --> 00:03:46.596
is that there is a
number-- So one caution.
00:03:46.596 --> 00:03:55.490
There's a number which I'll
call R, where R can be between 0
00:03:55.490 --> 00:03:57.350
and it can also be infinity.
00:03:57.350 --> 00:04:00.710
It's a number between 0
and infinity, inclusive,
00:04:00.710 --> 00:04:06.830
so that when the absolute
value of x is less than R.
00:04:06.830 --> 00:04:11.260
So when x is smaller than R
in size, the sum converges.
00:04:17.220 --> 00:04:21.170
This sum-- that sum
converges to a finite value.
00:04:21.170 --> 00:04:25.120
And when x is bigger
than R in absolute value,
00:04:25.120 --> 00:04:26.050
the sum diverges.
00:04:30.260 --> 00:04:32.165
This R is called the
radius of convergence.
00:04:42.960 --> 00:04:45.750
So we'll see some examples of
what the radius of convergence
00:04:45.750 --> 00:04:49.625
is in various powers series as
well, and how you find it also.
00:04:55.890 --> 00:04:57.710
But, let me go on and
give you a few more
00:04:57.710 --> 00:05:01.320
of the properties
about power series
00:05:01.320 --> 00:05:05.410
which I think that professor
Jerison talked about earlier.
00:05:05.410 --> 00:05:08.740
So one of them is there's
a radius of convergence.
00:05:08.740 --> 00:05:10.200
Here's another one.
00:05:15.760 --> 00:05:18.510
If you're inside of
the radius convergence,
00:05:18.510 --> 00:05:23.460
then the function has
all its derivatives,
00:05:23.460 --> 00:05:34.530
has all its derivatives,
just like a polynomial does.
00:05:34.530 --> 00:05:37.210
You can differentiate
it over and over again.
00:05:37.210 --> 00:05:41.300
And in terms of
those derivatives,
00:05:41.300 --> 00:05:46.880
the number a_n in
the power series
00:05:46.880 --> 00:05:52.000
can be expressed in terms of the
value of the derivative at 0.
00:05:52.000 --> 00:05:53.700
And this is called
Taylor's formula.
00:05:58.540 --> 00:06:02.390
So I'm saying that inside of
this radius of convergence,
00:06:02.390 --> 00:06:05.200
the function that we're
looking at, this f(x),
00:06:05.200 --> 00:06:10.710
can be written as the value of
the function at 0, that's a_0,
00:06:10.710 --> 00:06:13.830
plus the value of
the derivative.
00:06:13.830 --> 00:06:17.630
This bracket n means you
take the derivative n times.
00:06:17.630 --> 00:06:20.940
So when n is 1, you take
the derivative once at 0,
00:06:20.940 --> 00:06:25.100
divided by 1!, which is
!, and multiply it by x.
00:06:25.100 --> 00:06:27.630
That's the linear term
in the power series.
00:06:27.630 --> 00:06:30.260
And then the quadratic term is
you take the second derivative.
00:06:30.260 --> 00:06:33.740
Remember to divide
by 2!, which is 2.
00:06:33.740 --> 00:06:39.730
Multiply that by
x^2 and so on out.
00:06:39.730 --> 00:06:41.870
So, in terms-- So
the coefficients
00:06:41.870 --> 00:06:45.800
in the power series just record
the values of the derivatives
00:06:45.800 --> 00:06:48.080
of the function at x = 0.
00:06:48.080 --> 00:06:52.270
They can be computed
that way also.
00:06:52.270 --> 00:06:53.160
Let's see.
00:06:53.160 --> 00:06:55.760
I think that's the end
of my summary of things
00:06:55.760 --> 00:06:57.070
that he talked about.
00:06:57.070 --> 00:06:59.110
I think he did one
example, and I'll repeat
00:06:59.110 --> 00:07:04.680
that example of a power series.
00:07:04.680 --> 00:07:06.610
This example wasn't
due to David Jerison;
00:07:06.610 --> 00:07:07.860
it was due to Leonard Euler.
00:07:11.590 --> 00:07:15.490
It's the example of where the
function is the exponential
00:07:15.490 --> 00:07:16.210
function e^x.
00:07:19.280 --> 00:07:22.695
So, let's see.
00:07:22.695 --> 00:07:25.320
Let's compute what-- I will just
repeat for you the computation
00:07:25.320 --> 00:07:28.050
of the power series for
e^x, just because it's such
00:07:28.050 --> 00:07:30.142
an important thing to do.
00:07:30.142 --> 00:07:32.600
So, in order to do that, I have
to know what the derivative
00:07:32.600 --> 00:07:36.750
of e^x is, and what the
second derivative of e^x is,
00:07:36.750 --> 00:07:41.250
and so on, because that
comes into the Taylor formula
00:07:41.250 --> 00:07:42.690
for the coefficients.
00:07:42.690 --> 00:07:46.720
But we know what the derivative
of e^x is, it's just e^x again,
00:07:46.720 --> 00:07:49.310
and it's that way
all the way down.
00:07:49.310 --> 00:07:53.310
All the derivatives are
e^x over and over again.
00:07:53.310 --> 00:07:58.505
So when I evaluate this at x =
0, well, the value of e^x is 1,
00:07:58.505 --> 00:08:01.910
the value of e^x is 1 at x = 0.
00:08:01.910 --> 00:08:05.180
You get a value of
1 all the way down.
00:08:05.180 --> 00:08:10.090
So all these derivatives
at 0 have the value 1.
00:08:10.090 --> 00:08:12.790
And now, when I plug
into this formula,
00:08:12.790 --> 00:08:28.645
I find e^x is 1 plus 1*x
plus 1/2! x^2 plus 1/3! x^3,
00:08:28.645 --> 00:08:32.250
plus and so on.
00:08:32.250 --> 00:08:34.660
So all of these
numbers are 1, and all
00:08:34.660 --> 00:08:37.130
you wind up with is the
factorials in the denominators.
00:08:37.130 --> 00:08:38.870
That's the power series for e^x.
00:08:38.870 --> 00:08:42.050
This was a discovery of Leonhard
Euler in 1740 or something.
00:08:42.050 --> 00:08:42.842
Yes, Ma'am.
00:08:42.842 --> 00:08:45.102
AUDIENCE: When you're
writing out the power series,
00:08:45.102 --> 00:08:46.910
how far do you have
to write it out?
00:08:46.910 --> 00:08:49.243
PROFESSOR: How far do you
have to write the power series
00:08:49.243 --> 00:08:51.100
before it becomes well defined?
00:08:51.100 --> 00:08:54.340
Before it's a satisfactory
solution to an exam problem,
00:08:54.340 --> 00:08:58.100
I suppose, is another way
to phrase the question.
00:08:58.100 --> 00:09:00.920
Until you can see
what the pattern is.
00:09:00.920 --> 00:09:02.230
I can see what the pattern is.
00:09:02.230 --> 00:09:03.938
Is there anyone who's
in doubt about what
00:09:03.938 --> 00:09:07.662
the next term might be?
00:09:07.662 --> 00:09:09.120
Some people would
tell you that you
00:09:09.120 --> 00:09:11.900
have to write the
summation convention thing.
00:09:11.900 --> 00:09:13.862
Don't believe them.
00:09:13.862 --> 00:09:15.820
If you right out enough
terms to make it clear,
00:09:15.820 --> 00:09:17.020
that's good enough.
00:09:17.020 --> 00:09:18.987
OK?
00:09:18.987 --> 00:09:20.070
Is that an answer for you?
00:09:20.070 --> 00:09:22.990
AUDIENCE: Yes, Thank you.
00:09:22.990 --> 00:09:25.980
PROFESSOR: OK, so
that's a basic example.
00:09:25.980 --> 00:09:28.960
Let's do another basic
example of a power series.
00:09:28.960 --> 00:09:32.240
Oh yes, and by the way, whenever
you write out a power series,
00:09:32.240 --> 00:09:35.240
you should say what the
radius of convergence is.
00:09:35.240 --> 00:09:37.110
And for now, I will
just to tell you
00:09:37.110 --> 00:09:39.440
that the radius of convergence
of this power series
00:09:39.440 --> 00:09:42.870
is infinity; that
is, this sum always
00:09:42.870 --> 00:09:45.765
converges for any value of x.
00:09:45.765 --> 00:09:47.890
I'll say a little more
about that in a few minutes.
00:09:47.890 --> 00:09:49.780
Yeah?
00:09:49.780 --> 00:09:52.880
AUDIENCE: So which functions
can be written as power series?
00:09:52.880 --> 00:09:57.060
PROFESSOR: Which functions can
be written as power series?
00:09:57.060 --> 00:10:00.030
That's an excellent question.
00:10:00.030 --> 00:10:05.930
Any function that has
a reasonable expression
00:10:05.930 --> 00:10:08.679
can be written as
a power series.
00:10:08.679 --> 00:10:11.220
I'm not giving you a very good
answer because the true answer
00:10:11.220 --> 00:10:12.730
is a little bit complicated.
00:10:12.730 --> 00:10:14.470
But any of the
functions that occur
00:10:14.470 --> 00:10:18.420
in calculus like sines,
cosines, tangents, they all have
00:10:18.420 --> 00:10:21.495
power series expansions, OK?
00:10:21.495 --> 00:10:22.495
We'll see more examples.
00:10:25.830 --> 00:10:27.070
Let's do another example.
00:10:27.070 --> 00:10:30.130
Here's another example.
00:10:30.130 --> 00:10:31.520
I guess this was example one.
00:10:35.520 --> 00:10:42.140
So, this example, I think,
was due to Newton, not Euler.
00:10:42.140 --> 00:10:46.730
Let's find the power series
expansion of this function:
00:10:46.730 --> 00:10:48.200
1/(1+x).
00:10:48.200 --> 00:10:51.740
Well, I think that
somewhere along the line,
00:10:51.740 --> 00:10:56.137
you learned about the geometric
series which tells you
00:10:56.137 --> 00:10:58.220
that-- which tells you
what the answer to this is,
00:10:58.220 --> 00:11:00.190
and I'll just write it out.
00:11:00.190 --> 00:11:12.240
The geometric series tells
you that this function
00:11:12.240 --> 00:11:16.460
can be written as an
alternating sum of powers of x.
00:11:16.460 --> 00:11:18.810
You may wonder where
these minuses came from.
00:11:18.810 --> 00:11:21.100
Well, if you really think
about the geometric series,
00:11:21.100 --> 00:11:24.430
as you probably remembered,
there was a minus sign here,
00:11:24.430 --> 00:11:28.420
and that gets replaced
by these minus signs.
00:11:28.420 --> 00:11:31.810
I think maybe Jerison
talked about this also.
00:11:31.810 --> 00:11:34.640
Anyway, here's
another basic example.
00:11:34.640 --> 00:11:36.460
Remember what the
graph of this function
00:11:36.460 --> 00:11:41.404
looks like when x = -1.
00:11:41.404 --> 00:11:42.820
Then there's a
little problem here
00:11:42.820 --> 00:11:45.080
because the
denominator becomes 0,
00:11:45.080 --> 00:11:47.600
so the graph has a pole there.
00:11:47.600 --> 00:11:52.110
It goes up to
infinity at x = -1,
00:11:52.110 --> 00:11:57.620
and that's an indication that
the radius of convergence
00:11:57.620 --> 00:11:58.990
is not infinity.
00:11:58.990 --> 00:12:01.240
Because if you try to converge
to this infinite number
00:12:01.240 --> 00:12:04.930
by putting in x = -1, here,
you'll have a big problem.
00:12:04.930 --> 00:12:07.122
In fact, you see when
you put in x = -1,
00:12:07.122 --> 00:12:08.497
you keep getting
1 in every term,
00:12:08.497 --> 00:12:11.390
and it gets bigger and
bigger and does not converge.
00:12:11.390 --> 00:12:14.940
In this example, the
radius of convergence is 1.
00:12:18.570 --> 00:12:22.210
OK, so, let's do
a new example now.
00:12:22.210 --> 00:12:24.250
Oh, and by the way,
I should say you
00:12:24.250 --> 00:12:27.770
can calculate these numbers
using Taylor's formula.
00:12:27.770 --> 00:12:29.940
If you haven't seen
it, check it out.
00:12:29.940 --> 00:12:36.580
Calculate the iterated
derivatives of this function
00:12:36.580 --> 00:12:41.410
and plug in x = 0 and see
that you get +1, -1, +1, -1,
00:12:41.410 --> 00:12:41.930
and so on.
00:12:41.930 --> 00:12:42.706
Yes sir.
00:12:42.706 --> 00:12:44.636
AUDIENCE: For the
radius of convergence
00:12:44.636 --> 00:12:48.090
I see that if you do
-1 it'll blow out.
00:12:48.090 --> 00:12:50.740
If you put in 1 though, it
seems like it would be fine.
00:12:50.740 --> 00:12:52.550
PROFESSOR: The
questions is I can
00:12:52.550 --> 00:12:54.870
see that there's a
problem at x = -1,
00:12:54.870 --> 00:12:57.280
why is there also
a problem at x = 1
00:12:57.280 --> 00:12:59.090
where the graph is
perfectly smooth
00:12:59.090 --> 00:13:00.760
and innocuous and finite.
00:13:00.760 --> 00:13:04.490
That's another
excellent question.
00:13:04.490 --> 00:13:07.650
The problem is that if you
go off to a radius of 1
00:13:07.650 --> 00:13:11.530
in any direction and there's
a problem, that's it.
00:13:11.530 --> 00:13:13.530
That's what the radius
of convergence is.
00:13:13.530 --> 00:13:18.070
Here, what does happen
if I put an x = +1?
00:13:18.070 --> 00:13:20.490
So, let's look at
the partial sums.
00:13:20.490 --> 00:13:23.060
Do x = +1 in your mind here.
00:13:23.060 --> 00:13:29.190
So I'll get a partial sum 1,
then 0, and then 1, and then 0,
00:13:29.190 --> 00:13:29.907
and then 1.
00:13:29.907 --> 00:13:31.740
So even though it doesn't
go up to infinity,
00:13:31.740 --> 00:13:32.900
it still does not converge.
00:13:32.900 --> 00:13:35.500
AUDIENCE: And
anything in between?
00:13:35.500 --> 00:13:37.510
PROFESSOR: Any of
these other things
00:13:37.510 --> 00:13:41.330
will also fail to
converge in this example.
00:13:41.330 --> 00:13:43.685
Well, that's the only two
real numbers at the edge.
00:13:43.685 --> 00:13:44.185
Right?
00:13:46.940 --> 00:13:49.050
OK, let's do a
different example now.
00:13:49.050 --> 00:13:50.210
How about a trig function?
00:13:50.210 --> 00:13:50.793
The sine of x.
00:13:55.422 --> 00:14:01.810
I'm going to compute the power
series expansion for sin(x).
00:14:01.810 --> 00:14:04.400
and I'm going to do it
using Taylor's formula.
00:14:04.400 --> 00:14:06.310
So Taylor's formula
says that I have
00:14:06.310 --> 00:14:09.452
to start computing
derivatives of sin(x).
00:14:22.100 --> 00:14:25.280
Sounds like it's going
to be a lot of work.
00:14:25.280 --> 00:14:28.005
Let's see, the derivative
of the sine is the cosine.
00:14:30.870 --> 00:14:32.910
And the derivative
of the cosine,
00:14:32.910 --> 00:14:36.530
that's the second derivative
of the sine, is what?
00:14:36.530 --> 00:14:40.270
Remember the minus,
it's -sin(x).
00:14:40.270 --> 00:14:43.180
OK, now I want to take the third
derivative of the sine, which
00:14:43.180 --> 00:14:45.680
is the derivative
of sine prime prime,
00:14:45.680 --> 00:14:47.760
so it's the derivative of this.
00:14:47.760 --> 00:14:49.840
And we just decided
the derivative of sine
00:14:49.840 --> 00:14:52.270
is cosine, so I
get cosine, but I
00:14:52.270 --> 00:14:53.730
have this minus sign in front.
00:14:56.660 --> 00:14:58.710
And now I want to
differentiate again,
00:14:58.710 --> 00:15:01.640
so the cosine
becomes a minus sine,
00:15:01.640 --> 00:15:08.520
and that sign cancels with this
minus sign to give me sin(x).
00:15:08.520 --> 00:15:10.102
You follow that?
00:15:10.102 --> 00:15:13.660
It's a lot of -1's
canceling out there.
00:15:13.660 --> 00:15:17.290
So, all of a sudden, I'm
right back where I started;
00:15:17.290 --> 00:15:21.610
these two are the same and the
pattern will now repeat forever
00:15:21.610 --> 00:15:22.780
and ever.
00:15:22.780 --> 00:15:24.440
Higher and higher
derivatives of sines
00:15:24.440 --> 00:15:28.830
are just plus or minus
sines and cosines.
00:15:28.830 --> 00:15:34.300
Now Taylor's formula says I
should now substitute x = 0
00:15:34.300 --> 00:15:37.580
into this and see what
happens, so let's do that.
00:15:37.580 --> 00:15:43.240
When x is equals to 0, the
sine is 0 and the cosine is 1.
00:15:43.240 --> 00:15:47.410
The sine is 0, so
minus 0 is also 0.
00:15:47.410 --> 00:15:51.070
The cosine is 1, but
now there's a minus one,
00:15:51.070 --> 00:15:53.720
and now I'm back
where I started,
00:15:53.720 --> 00:15:58.760
and so the pattern will repeat.
00:15:58.760 --> 00:16:00.670
OK, so the values
of the derivatives
00:16:00.670 --> 00:16:03.630
are all zeros and
plus and minus ones
00:16:03.630 --> 00:16:07.420
and they go through that
pattern, four-fold periodicity,
00:16:07.420 --> 00:16:09.670
over and over again.
00:16:09.670 --> 00:16:13.277
And so we can write
out what sin(x)
00:16:13.277 --> 00:16:15.410
is using Taylor's formula,
using this formula.
00:16:18.000 --> 00:16:21.770
So I put in the value
at 0 which is 0, then
00:16:21.770 --> 00:16:27.620
I put in the derivative
which is 1, multiplied by x.
00:16:27.620 --> 00:16:32.180
Then, I have the second
derivative divided by 2!,
00:16:32.180 --> 00:16:35.150
but the second
derivative at 0 is 0.
00:16:35.150 --> 00:16:38.280
So I'm going to
drop that term out.
00:16:38.280 --> 00:16:41.365
Now I have the third
derivative which is -1.
00:16:43.930 --> 00:16:45.550
And remember the 3!
00:16:45.550 --> 00:16:46.790
in the denominator.
00:16:46.790 --> 00:16:50.050
That's the coefficient of x^3.
00:16:50.050 --> 00:16:51.680
What's the fourth derivative?
00:16:51.680 --> 00:16:54.400
Well, here we are, it's
on the board, it's 0.
00:16:54.400 --> 00:16:58.150
So I drop that term out
go up to the fifth term,
00:16:58.150 --> 00:16:59.830
the fifth power of x.
00:16:59.830 --> 00:17:02.260
Its derivative is now 1.
00:17:02.260 --> 00:17:06.750
We've gone through the pattern,
we're back at +1 as the value
00:17:06.750 --> 00:17:13.180
of the iterated derivative,
so now I get 1/5! x^5.
00:17:13.180 --> 00:17:15.720
Now, you tell me, have we
done enough terms to see
00:17:15.720 --> 00:17:17.900
what the pattern is?
00:17:17.900 --> 00:17:22.170
I guess the next
term will be a -1/7!
00:17:22.170 --> 00:17:23.750
x^7, and so on.
00:17:23.750 --> 00:17:28.160
Let me write this out
again just so we have it.
00:17:28.160 --> 00:17:30.830
x^3 / 3!-- So it's
x minus x^3 / 3!
00:17:30.830 --> 00:17:31.680
plus x^5 / 5!.
00:17:34.580 --> 00:17:38.740
You guessed it, and so on.
00:17:38.740 --> 00:17:40.270
That's the power
series expansion
00:17:40.270 --> 00:17:43.935
for the sine of x, OK?
00:17:46.950 --> 00:17:49.700
And so, the sign alternate,
and these denominators
00:17:49.700 --> 00:17:52.250
get very big, don't they?
00:17:52.250 --> 00:17:54.410
Exponentials grow very fast.
00:17:54.410 --> 00:17:55.830
Let me make a remark.
00:17:55.830 --> 00:17:58.800
R is infinity here.
00:17:58.800 --> 00:18:01.580
The radius of convergence
of this power series
00:18:01.580 --> 00:18:03.424
again is infinity, and
let me just say why.
00:18:03.424 --> 00:18:13.110
The reason is that the general
term is going to be like
00:18:13.110 --> 00:18:14.100
x^(2n+1) / (2n+1)!.
00:18:18.330 --> 00:18:21.900
An odd number I can
write as 2n + 1.
00:18:21.900 --> 00:18:24.608
And what I want to
say is that the size
00:18:24.608 --> 00:18:30.480
of this, what happens
to the size of this as n
00:18:30.480 --> 00:18:33.989
goes to infinity?
00:18:33.989 --> 00:18:35.280
So let's just think about this.
00:18:35.280 --> 00:18:38.270
For a fixed x, let's
fix the number x.
00:18:38.270 --> 00:18:41.260
Look at powers of x and
think about the size
00:18:41.260 --> 00:18:45.682
of this expression when
n gets to be large.
00:18:45.682 --> 00:18:47.140
So let's just do
that for a second.
00:18:47.140 --> 00:18:54.420
So, x^(2n+1) / (2n+1)!, I
can write out like this.
00:18:54.420 --> 00:19:03.010
It's x / 1 times x / 2
-- sorry -- times x / 3,
00:19:03.010 --> 00:19:09.250
times x / (2n+1).
00:19:09.250 --> 00:19:13.220
I've multiplied x by itself
2n+1 times in the numerator,
00:19:13.220 --> 00:19:15.680
and I've multiplied
the numbers 1, 2, 3, 4,
00:19:15.680 --> 00:19:18.477
and so on, by each other
in the denominator,
00:19:18.477 --> 00:19:19.810
and that gives me the factorial.
00:19:19.810 --> 00:19:22.330
So I've just written
this out like this.
00:19:22.330 --> 00:19:26.820
Now x is fixed, so maybe
it's a million, OK?
00:19:26.820 --> 00:19:28.700
It's big, but fixed.
00:19:28.700 --> 00:19:30.634
What happens to these numbers?
00:19:30.634 --> 00:19:32.050
Well at first,
they're pretty big.
00:19:32.050 --> 00:19:34.560
This is 1,000,000 / 2,
this is 1,000,000 / 3.
00:19:34.560 --> 00:19:38.820
But when n gets to be--
Maybe if n is 1,000,000,
00:19:38.820 --> 00:19:41.180
then this is about 1/2.
00:19:41.180 --> 00:19:48.320
If n is a billion, then this
is about 1/2,000, right?
00:19:48.320 --> 00:19:50.530
The denominators keep
getting bigger and bigger,
00:19:50.530 --> 00:19:54.470
but the numerators stay
the same; they're always x.
00:19:54.470 --> 00:19:57.590
So when I take the product,
if I go far enough out,
00:19:57.590 --> 00:20:00.070
I'm going to be multiplying,
by very, very small numbers
00:20:00.070 --> 00:20:01.730
and more and more of them.
00:20:01.730 --> 00:20:05.980
And so no matter what
x is, these numbers
00:20:05.980 --> 00:20:07.220
will converge to 0.
00:20:07.220 --> 00:20:11.550
They'll get smaller and
smaller as x gets to be bigger.
00:20:11.550 --> 00:20:16.590
That's the sign that x is inside
of the radius of convergence.
00:20:16.590 --> 00:20:21.360
This is the sign for
you that this series
00:20:21.360 --> 00:20:23.780
converges for that value of x.
00:20:23.780 --> 00:20:33.600
And because I could do
this for any x, this works.
00:20:33.600 --> 00:20:40.260
This convergence to
0 for any fixed x.
00:20:40.260 --> 00:20:43.330
That's what tells
you that you can
00:20:43.330 --> 00:20:46.060
take-- that the radius of
convergence is infinity.
00:20:46.060 --> 00:20:49.575
Because in the
formula, in the fact,
00:20:49.575 --> 00:20:53.700
in this property that
the radius of convergence
00:20:53.700 --> 00:20:56.340
talks about, if R is
equal to infinity,
00:20:56.340 --> 00:20:58.160
this is no condition on x.
00:20:58.160 --> 00:21:02.250
Every number is less than
infinity in absolute value.
00:21:02.250 --> 00:21:05.930
So if this convergence
to 0 of the general term
00:21:05.930 --> 00:21:10.320
works for every x, then radius
of convergence is infinity.
00:21:10.320 --> 00:21:11.700
Well that was kind
of fast, but I
00:21:11.700 --> 00:21:13.790
think that you've heard
something about that
00:21:13.790 --> 00:21:16.020
earlier as well.
00:21:16.020 --> 00:21:19.140
Anyway, so we've got the
sine function, a new function
00:21:19.140 --> 00:21:20.880
with its own power series.
00:21:20.880 --> 00:21:23.730
It's a way of computing sin(x).
00:21:23.730 --> 00:21:26.660
If you take enough
terms you'll get
00:21:26.660 --> 00:21:28.560
a good evaluation of sin(x).
00:21:28.560 --> 00:21:30.000
for any x.
00:21:30.000 --> 00:21:32.200
This tells you a lot
about the function sin(x)
00:21:32.200 --> 00:21:33.750
but not everything at all.
00:21:33.750 --> 00:21:36.590
For example, from
this formula, it's
00:21:36.590 --> 00:21:39.745
very hard to see that the
sine of x is periodic.
00:21:39.745 --> 00:21:41.930
It's not obvious at all.
00:21:41.930 --> 00:21:44.090
Somewhere hidden away
in this expression
00:21:44.090 --> 00:21:47.400
is the number pi, the
half of the period.
00:21:47.400 --> 00:21:51.100
But that's not clear from
the power series at all.
00:21:51.100 --> 00:21:53.320
So the power series are
very good for some things,
00:21:53.320 --> 00:21:55.510
but they hide other
properties of functions.
00:21:58.150 --> 00:22:00.830
Well, so I want to spend
a few minutes telling you
00:22:00.830 --> 00:22:04.350
about what you can do
with a power series,
00:22:04.350 --> 00:22:07.620
once you have one, to get new
power series, so new power
00:22:07.620 --> 00:22:08.530
series from old.
00:22:18.300 --> 00:22:25.490
And this is also called
operations on power series.
00:22:25.490 --> 00:22:27.989
So what are the things that
we can do to a power series?
00:22:27.989 --> 00:22:29.905
Well one of the things
you can do is multiply.
00:22:33.990 --> 00:22:37.310
So, for example, what if
I want to compute a power
00:22:37.310 --> 00:22:40.970
series for x sin(x)?
00:22:40.970 --> 00:22:44.160
Well I have a power series
for sin(x), I just did it.
00:22:44.160 --> 00:22:45.920
How about a power series for x?
00:22:48.910 --> 00:22:51.480
Actually, I did that here too.
00:22:51.480 --> 00:22:55.120
The function x is a
very simple polynomial.
00:22:55.120 --> 00:22:58.330
It's a polynomial where
that's 0, a_1 is 1,
00:22:58.330 --> 00:23:00.930
and all the other
coefficients are 0.
00:23:00.930 --> 00:23:04.870
So x itself is a power
series, a very simple one.
00:23:04.870 --> 00:23:08.317
sin(x) is a powers series.
00:23:08.317 --> 00:23:09.900
And what I want to
encourage you to do
00:23:09.900 --> 00:23:12.720
is treat power series
just like polynomials
00:23:12.720 --> 00:23:14.330
and multiply them together.
00:23:14.330 --> 00:23:16.960
We'll see other operations too.
00:23:16.960 --> 00:23:20.860
So, to compute the power series
for x sin(x), of I just take
00:23:20.860 --> 00:23:24.400
this one and multiply it by x.
00:23:24.400 --> 00:23:26.650
So let's see if I
can do that right.
00:23:26.650 --> 00:23:30.050
It distributes through:
x^2 minus x^4 / 3!
00:23:33.000 --> 00:23:42.190
plus x^6 / 5!, and so on.
00:23:42.190 --> 00:23:44.180
And again, the
radius of convergence
00:23:44.180 --> 00:23:47.130
is going to be the smaller of
the two radii of convergence
00:23:47.130 --> 00:23:48.220
here.
00:23:48.220 --> 00:23:51.840
So it's R equals
infinity in this case.
00:23:51.840 --> 00:23:54.010
OK, you can multiply
power series together.
00:23:54.010 --> 00:23:56.910
It can be a pain if the
power series are very long,
00:23:56.910 --> 00:24:01.800
but if one of them is
x, it's pretty simple.
00:24:01.800 --> 00:24:06.040
OK, that's one thing I can do.
00:24:06.040 --> 00:24:08.634
Notice something by the way.
00:24:08.634 --> 00:24:10.175
You know that even
and odd functions?
00:24:13.180 --> 00:24:17.390
So, sine is an odd function,
x is an odd function,
00:24:17.390 --> 00:24:20.570
the product of two odd
functions is an even function.
00:24:20.570 --> 00:24:24.070
And that's reflected in the fact
that all the powers that occur
00:24:24.070 --> 00:24:26.790
in the power series are even.
00:24:26.790 --> 00:24:30.640
For an odd function, like the
sine, all the powers that occur
00:24:30.640 --> 00:24:32.589
are odd powers of x.
00:24:32.589 --> 00:24:33.380
That's always true.
00:24:37.510 --> 00:24:38.969
OK, we can multiply.
00:24:38.969 --> 00:24:40.010
I can also differentiate.
00:24:48.660 --> 00:24:56.950
So let's just do a
case of that, and use
00:24:56.950 --> 00:24:59.240
the process of
differentiation to find out
00:24:59.240 --> 00:25:03.580
what the power
series for cos(x) is
00:25:03.580 --> 00:25:06.560
by writing the cos(x) as
the derivative of the sine
00:25:06.560 --> 00:25:09.010
and differentiating
term by term.
00:25:09.010 --> 00:25:11.360
So, I'll take this
expression for the power
00:25:11.360 --> 00:25:14.200
series of the sine and
differentiate it term by term,
00:25:14.200 --> 00:25:18.210
and I'll get the power
series for cosine.
00:25:18.210 --> 00:25:19.030
So, let's see.
00:25:19.030 --> 00:25:22.510
The derivative of x is one.
00:25:22.510 --> 00:25:27.110
Now, the derivative of x^3 is
3x^2, and then there's a 3!
00:25:27.110 --> 00:25:28.910
in the denominator.
00:25:28.910 --> 00:25:34.440
And the derivative of x^5
5x^4, and there's a 5!
00:25:34.440 --> 00:25:38.680
in the denominator,
and so on and so on.
00:25:38.680 --> 00:25:40.950
And now some
cancellation happens.
00:25:40.950 --> 00:25:45.960
So this is 1 minus, well, the
3 cancels with the last factor
00:25:45.960 --> 00:25:48.730
in this 3 factorial
and leaves you with 2!.
00:25:52.460 --> 00:25:56.040
And the 5 cancels with the
last factor in the 5 factorial
00:25:56.040 --> 00:25:58.129
and leaves you with a 4!
00:25:58.129 --> 00:25:58.920
in the denominator.
00:26:01.570 --> 00:26:04.710
And so there you go, there's
the power series expansion
00:26:04.710 --> 00:26:05.980
for the cosine.
00:26:05.980 --> 00:26:07.970
It's got all even powers of x.
00:26:07.970 --> 00:26:12.720
They alternate, and you have
factorials in the denominator.
00:26:12.720 --> 00:26:15.510
And of course, you could
derive that expression
00:26:15.510 --> 00:26:19.130
by using Taylor's formula, by
the same kind of calculation
00:26:19.130 --> 00:26:22.290
you did here, taking higher
and higher derivatives
00:26:22.290 --> 00:26:22.970
of the cosine.
00:26:22.970 --> 00:26:25.720
You get the same
periodic pattern
00:26:25.720 --> 00:26:30.080
of derivatives and values
of derivatives at x = 0.
00:26:30.080 --> 00:26:33.200
But here's a cleaner way to
do it, simpler way to do it,
00:26:33.200 --> 00:26:36.830
because we already knew
the derivative of the sine.
00:26:36.830 --> 00:26:39.630
When you differentiate, you keep
the same radius of convergence.
00:26:44.420 --> 00:26:49.320
OK, so we can
multiply, I can add too
00:26:49.320 --> 00:26:52.400
and multiply by a
constant, things like that.
00:26:52.400 --> 00:26:54.280
How about integrating?
00:26:54.280 --> 00:26:56.580
That's what half of this
course was about isn't it?
00:26:56.580 --> 00:26:58.550
So, let's integrate something.
00:27:07.210 --> 00:27:15.960
So, the integration I'm
going to do is this one:
00:27:15.960 --> 00:27:20.160
the integral from 0
to x of dt / (1+x).
00:27:20.160 --> 00:27:21.980
What is that integral
as a function?
00:27:28.360 --> 00:27:32.060
So, when I find the
anti-derivative of this,
00:27:32.060 --> 00:27:39.660
I get ln(1+t), and then when
I evaluate that at t = x,
00:27:39.660 --> 00:27:42.600
I get ln(1+x).
00:27:42.600 --> 00:27:48.510
And when I evaluate the natural
log at 0, I get the ln 1,
00:27:48.510 --> 00:27:55.070
which is 0, so this
is what you get, OK?
00:27:55.070 --> 00:28:05.690
This is really valid, by the
way, for x bigger than -1.
00:28:05.690 --> 00:28:09.060
But you don't want to think
about this quite like this
00:28:09.060 --> 00:28:10.240
when x is smaller than that.
00:28:13.770 --> 00:28:18.900
Now, I'm going to try to apply
power series methods here
00:28:18.900 --> 00:28:22.660
and find-- use this integral
to find a power series
00:28:22.660 --> 00:28:28.230
for the natural log, and I'll
do it by plugging into this
00:28:28.230 --> 00:28:34.330
expression what the power
series for 1/(1+t) was.
00:28:34.330 --> 00:28:36.830
And I know what that is because
I wrote it down on the board
00:28:36.830 --> 00:28:38.330
up here.
00:28:38.330 --> 00:28:40.940
Change the variable
from x to t there,
00:28:40.940 --> 00:28:49.410
and so 1/(1+t) is 1 minus t
plus t^2 minus t^3, and so on.
00:28:52.620 --> 00:28:55.196
So that's the thing in the
inside of the integral,
00:28:55.196 --> 00:29:01.710
and now it's legal to
integrate that term by term,
00:29:01.710 --> 00:29:03.517
so let's do that.
00:29:03.517 --> 00:29:05.350
I'm going to get something
which I will then
00:29:05.350 --> 00:29:09.230
evaluate at x and at 0.
00:29:09.230 --> 00:29:14.093
So, when I integrate 1 I get
x, and when I integrate t,
00:29:14.093 --> 00:29:14.593
I get t.
00:29:14.593 --> 00:29:16.829
I'm sorry.
00:29:16.829 --> 00:29:29.100
When I integrate t, I get t^2
/ 2, and t^2 gives me t^3 / 3,
00:29:29.100 --> 00:29:29.970
and so on and so on.
00:29:32.520 --> 00:29:36.350
And then, when I
put in t = x, well,
00:29:36.350 --> 00:29:40.350
I just replace all the t's by
x's, and when I put in t = 0,
00:29:40.350 --> 00:29:41.780
I get 0.
00:29:41.780 --> 00:29:43.950
So this equals x.
00:29:43.950 --> 00:29:55.170
So, I've discovered that ln(1+x)
is x minus x^2 / 2 plus x^3 / 3
00:29:55.170 --> 00:30:02.020
minus x^4 / 4, and
so on and so on.
00:30:02.020 --> 00:30:04.720
There's the power series
expansion for ln(1+x).
00:30:07.800 --> 00:30:10.280
And because I began
with a power series
00:30:10.280 --> 00:30:13.030
whose radius of
convergence was just 1,
00:30:13.030 --> 00:30:15.930
I began with this power
series, the radius
00:30:15.930 --> 00:30:18.150
of convergence of this
is also going to be 1.
00:30:22.200 --> 00:30:25.670
Also, because this function,
as I just pointed out,
00:30:25.670 --> 00:30:29.080
this function goes bad when
x becomes less than -1,
00:30:29.080 --> 00:30:32.160
so some problem happens,
and that's reflected
00:30:32.160 --> 00:30:35.590
in the radius of convergence.
00:30:35.590 --> 00:30:36.750
Cool.
00:30:36.750 --> 00:30:41.110
So, you can integrate.
00:30:41.110 --> 00:30:45.570
That is the correct power series
expansion for the ln(1+x),
00:30:45.570 --> 00:30:49.340
and another victory of Euler's
was to use this kind of power
00:30:49.340 --> 00:30:52.270
series expansion to calculate
natural logarithms in a much
00:30:52.270 --> 00:30:54.350
more efficient way than
people had done before.
00:30:57.850 --> 00:31:08.380
OK, one more property, I think.
00:31:12.920 --> 00:31:17.080
What are we at here, 3?
00:31:17.080 --> 00:31:18.550
4.
00:31:18.550 --> 00:31:19.050
Substitute.
00:31:25.410 --> 00:31:28.669
Very appropriate for me
as a substitute teacher
00:31:28.669 --> 00:31:29.960
to tell you about substitution.
00:31:32.810 --> 00:31:35.480
So I'm going to try to find
the power series expansion
00:31:35.480 --> 00:31:36.385
of e^(-t^2).
00:31:36.385 --> 00:31:36.885
OK?
00:31:41.740 --> 00:31:45.190
And the way I'll do that is
by taking the power series
00:31:45.190 --> 00:31:50.050
expansion for e^x,
which we have up there,
00:31:50.050 --> 00:32:01.630
and make the substitution x =
-t^2 in the expansion for e^x.
00:32:01.630 --> 00:32:03.125
Did you have a question?
00:32:03.125 --> 00:32:04.625
AUDIENCE: Well,
it's just concerning
00:32:04.625 --> 00:32:07.624
the radius of convergence.
00:32:07.624 --> 00:32:11.660
You can't define x so that is
always positive, and if so,
00:32:11.660 --> 00:32:14.360
it wouldn't have a radius
of convergence, right?
00:32:14.360 --> 00:32:20.740
PROFESSOR: Like I say, again the
worry is this ln(1+x) function
00:32:20.740 --> 00:32:24.120
is perfectly well
behaved for large x.
00:32:24.120 --> 00:32:27.720
Why does the power series
fail to converge for large x?
00:32:27.720 --> 00:32:30.260
Well suppose that
x is bigger than 1,
00:32:30.260 --> 00:32:32.200
then here you get
bigger and bigger powers
00:32:32.200 --> 00:32:35.160
of x, which will
grow to infinity,
00:32:35.160 --> 00:32:40.330
and they grow large faster
than the numbers 2, 3, 4, 5, 6.
00:32:40.330 --> 00:32:45.850
They grow exponentially, and
these just grow linearly.
00:32:45.850 --> 00:32:49.430
So, again, the general term,
when x is bigger than one,
00:32:49.430 --> 00:32:51.600
the general term will
go off to infinity,
00:32:51.600 --> 00:32:53.840
even though the function
that you're talking about,
00:32:53.840 --> 00:32:56.750
log of net of 1 plus
x is perfectly good.
00:32:56.750 --> 00:33:01.615
So the power series is not
good outside of the radius
00:33:01.615 --> 00:33:02.240
of convergence.
00:33:02.240 --> 00:33:04.650
It's just a fact of life.
00:33:04.650 --> 00:33:05.150
Yes?
00:33:05.150 --> 00:33:06.025
AUDIENCE: [INAUDIBLE]
00:33:18.332 --> 00:33:20.290
PROFESSOR: I'd rather--
talk to me after class.
00:33:20.290 --> 00:33:22.620
The question is why is
it the smaller of the two
00:33:22.620 --> 00:33:24.110
radii of convergence?
00:33:24.110 --> 00:33:30.050
The basic answer
is, well, you can't
00:33:30.050 --> 00:33:33.370
expect it to be bigger than that
smaller one, because the power
00:33:33.370 --> 00:33:34.790
series only gives
you information
00:33:34.790 --> 00:33:37.020
inside of that range
about the function, so.
00:33:37.020 --> 00:33:37.895
AUDIENCE: [INAUDIBLE]
00:33:41.010 --> 00:33:43.390
PROFESSOR: Well, in this
case, both of the radii
00:33:43.390 --> 00:33:44.890
of convergence are infinity.
00:33:44.890 --> 00:33:48.110
x has radius of convergence
infinity for sure,
00:33:48.110 --> 00:33:49.260
and sin(x) does too.
00:33:49.260 --> 00:33:52.140
So you get infinity
in that case, OK?
00:33:54.850 --> 00:33:58.162
OK, let's just do
this, and then I'm
00:33:58.162 --> 00:33:59.620
going to integrate
this and that'll
00:33:59.620 --> 00:34:03.830
be the end of what I
have time for today.
00:34:03.830 --> 00:34:05.880
So what's the power
series expansion for this?
00:34:05.880 --> 00:34:07.490
The power series
expansion of this
00:34:07.490 --> 00:34:11.160
is going to be a
function of t, right,
00:34:11.160 --> 00:34:13.530
because the variable here is t.
00:34:13.530 --> 00:34:19.360
I get it by taking my expansion
for e^x and putting in what x
00:34:19.360 --> 00:34:20.460
is in terms of t.
00:34:30.210 --> 00:34:32.180
Whoops!
00:34:32.180 --> 00:34:36.700
And so on and so on.
00:34:36.700 --> 00:34:42.100
I just put in -t^2 in place of
x there in the series expansion
00:34:42.100 --> 00:34:44.120
for e^x.
00:34:44.120 --> 00:34:47.910
I can work this out
a little bit better.
00:34:47.910 --> 00:34:49.030
-t^2 is what it is.
00:34:49.030 --> 00:34:53.760
This is going to give me a t^4
and the minus squared is going
00:34:53.760 --> 00:34:55.790
to give me a plus,
so I get t^4 / 2!.
00:34:58.730 --> 00:35:05.750
Then I get (-t)^3, so there'll
be a minus sign and a t^6
00:35:05.750 --> 00:35:08.190
and the denominator 3!.
00:35:08.190 --> 00:35:10.300
So the signs are
going to alternate,
00:35:10.300 --> 00:35:13.430
the powers are all even,
and the denominators
00:35:13.430 --> 00:35:15.380
are these factorials.
00:35:20.160 --> 00:35:23.950
Several times as this
course has gone on,
00:35:23.950 --> 00:35:27.216
the error function has
made an appearance.
00:35:27.216 --> 00:35:31.790
The error function was, I guess
it gets normalized by putting
00:35:31.790 --> 00:35:41.550
a 2 over the square
root of pi in front,
00:35:41.550 --> 00:35:46.830
and it's the integral of
e^(-t^2) dt from 0 to x.
00:35:46.830 --> 00:35:54.700
And this normalization
is here because as x
00:35:54.700 --> 00:36:01.300
gets to be large
the value becomes 1.
00:36:01.300 --> 00:36:04.305
So this error function is
very important in the theory
00:36:04.305 --> 00:36:06.080
of probability.
00:36:06.080 --> 00:36:09.120
And I think you calculated
this fact at some point
00:36:09.120 --> 00:36:12.236
in the course.
00:36:12.236 --> 00:36:14.960
So the standard definition of
the error function, you put a 2
00:36:14.960 --> 00:36:16.460
over the square
root of pi in front.
00:36:16.460 --> 00:36:18.425
Let's calculate its
power series expansion.
00:36:21.320 --> 00:36:23.220
So there's a 2 over
the square root of pi
00:36:23.220 --> 00:36:27.280
that hurts nobody
here in the front.
00:36:27.280 --> 00:36:30.530
And now I want to
integrate e^(-t^2),
00:36:30.530 --> 00:36:34.010
and I'm going to use this
power series expansion for that
00:36:34.010 --> 00:36:36.350
to see what you get.
00:36:36.350 --> 00:36:38.800
So I'm just going to
write this out I think.
00:36:38.800 --> 00:36:41.340
I did it out carefully in
another example over there,
00:36:41.340 --> 00:36:43.100
so I'll do it a
little quicker now.
00:36:43.100 --> 00:36:45.606
Integrate this term
by term, you're
00:36:45.606 --> 00:36:47.730
just integrating powers of
t so it's pretty simple,
00:36:47.730 --> 00:36:51.830
so I get-- and then I'm
evaluating at x and then at 0.
00:36:51.830 --> 00:37:03.690
So I get x minus x^3 /
3, plus x^5 / (5*2!),
00:37:03.690 --> 00:37:07.790
5 from integrating
the t^4, and the 2!
00:37:07.790 --> 00:37:11.390
from this denominator
that we already had.
00:37:11.390 --> 00:37:18.040
And then there's a -x^7
/ (7*3!), and plus,
00:37:18.040 --> 00:37:21.620
and so on, and you can imagine
how they go on from there.
00:37:24.490 --> 00:37:27.500
I guess to get this
exactly in the form
00:37:27.500 --> 00:37:32.510
that we began talking about,
I should multiply through.
00:37:32.510 --> 00:37:35.810
So the coefficient of x is 2
over the square root of pi,
00:37:35.810 --> 00:37:39.465
and the coefficient of x^3 is
-2 over 3 times the square root
00:37:39.465 --> 00:37:41.012
of pi, and so on.
00:37:41.012 --> 00:37:43.470
But this is a perfectly good
way to write this power series
00:37:43.470 --> 00:37:45.630
expansion as well.
00:37:45.630 --> 00:37:49.020
And, this is a very good way to
compute the value of the error
00:37:49.020 --> 00:37:49.570
function.
00:37:49.570 --> 00:37:53.210
It's a new function
in our experience.
00:37:53.210 --> 00:37:55.470
Your calculator
probably calculates it,
00:37:55.470 --> 00:37:58.710
and your calculator probably
does it by this method.
00:38:01.270 --> 00:38:07.480
OK, so that's my sermon
on examples of things
00:38:07.480 --> 00:38:10.260
you can do with power series.
00:38:10.260 --> 00:38:13.870
So, we're going to do the
CEG thing in just a minute.
00:38:13.870 --> 00:38:17.740
Professor Jerison wanted
me to make an ad for 18.02.
00:38:17.740 --> 00:38:20.620
Just in case you were thinking
of not taking it next term,
00:38:20.620 --> 00:38:21.980
you really should take it.
00:38:21.980 --> 00:38:24.810
It will put a lot of
things in this course
00:38:24.810 --> 00:38:26.720
into context, for one thing.
00:38:26.720 --> 00:38:29.030
It's about vector
calculus and so on.
00:38:29.030 --> 00:38:32.190
So you'll learn about
vectors and things like that.
00:38:32.190 --> 00:38:34.701
But it comes back and
explains some things
00:38:34.701 --> 00:38:36.700
in this course that might
have been a little bit
00:38:36.700 --> 00:38:43.460
strange, like these strange
formulas for the product
00:38:43.460 --> 00:38:48.710
rule and the quotient rule and
the sort of random formulas.
00:38:48.710 --> 00:38:50.790
Well, one of the things
you learn in 18.02
00:38:50.790 --> 00:38:54.560
is that they're all special
cases of the chain rule.
00:38:54.560 --> 00:38:57.480
And just to drive
that point home,
00:38:57.480 --> 00:39:02.530
he wanted me to show you
this poem of his that
00:39:02.530 --> 00:39:06.000
really drives the points
home forcefully, I think.