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PROFESSOR: Last time we
left off with a question
00:00:24.520 --> 00:00:26.185
having to do with
playing with blocks.
00:00:26.185 --> 00:00:30.450
And this is supposed to give us
a visceral feel for something
00:00:30.450 --> 00:00:32.820
anyway, having to
do with series.
00:00:32.820 --> 00:00:35.650
And the question was whether
I could stack these blocks,
00:00:35.650 --> 00:00:39.460
build up a stack so that--
I'm going to try here.
00:00:39.460 --> 00:00:42.920
I'm already off
balance here, see.
00:00:42.920 --> 00:00:46.150
The question is
can I build this so
00:00:46.150 --> 00:00:51.350
that the-- let's
draw a picture of it,
00:00:51.350 --> 00:00:54.870
so that the first
block is like this.
00:00:54.870 --> 00:00:56.567
The next block is like this.
00:00:56.567 --> 00:00:58.150
And maybe the next
block is like this.
00:00:58.150 --> 00:01:00.240
And notice there
is no visible means
00:01:00.240 --> 00:01:02.410
of support for this block.
00:01:02.410 --> 00:01:08.310
It's completely to the
left of the first block.
00:01:08.310 --> 00:01:11.055
And the question is,
will this fall down?
00:01:13.730 --> 00:01:21.610
Or at least, or more precisely,
eventually we'll ask,
00:01:21.610 --> 00:01:24.650
you know, how far can we go?
00:01:24.650 --> 00:01:28.760
Now before you
answer this question,
00:01:28.760 --> 00:01:30.900
the claim is that
this is a kind of
00:01:30.900 --> 00:01:33.480
a natural, physical
question, which
00:01:33.480 --> 00:01:37.330
involves some important answer.
00:01:37.330 --> 00:01:40.290
No matter whether the answer
is you can do it or you can't.
00:01:40.290 --> 00:01:42.090
So this is a good
kind of math question
00:01:42.090 --> 00:01:44.440
where no matter what the
answer is, when you figure out
00:01:44.440 --> 00:01:46.850
the answer, you're going to
get something interesting out
00:01:46.850 --> 00:01:47.350
of it.
00:01:47.350 --> 00:01:48.766
Because they're
two possibilities.
00:01:48.766 --> 00:01:53.055
Either there is a limit to how
far to the left we can go --
00:01:53.055 --> 00:01:56.400
in which case that's a
very interesting number --
00:01:56.400 --> 00:01:58.810
or else there is no limit.
00:01:58.810 --> 00:02:00.370
You can go arbitrarily far.
00:02:00.370 --> 00:02:03.240
And that's also
interesting and curious.
00:02:03.240 --> 00:02:05.400
And that's the difference
between convergence
00:02:05.400 --> 00:02:08.190
and divergence,
the thing that we
00:02:08.190 --> 00:02:11.380
were talking about up to
now concerning series.
00:02:11.380 --> 00:02:13.890
So my first question
is, do you think
00:02:13.890 --> 00:02:19.810
that I can get it so that
this thing doesn't fall down
00:02:19.810 --> 00:02:24.020
with-- well you see I have
about eight blocks here or so.
00:02:24.020 --> 00:02:25.570
So you can vote now.
00:02:25.570 --> 00:02:27.750
How many in favor
that I can succeed
00:02:27.750 --> 00:02:30.450
in doing this sort of thing with
maybe more than three blocks.
00:02:30.450 --> 00:02:32.040
How many in favor?
00:02:32.040 --> 00:02:33.860
All right somebody
is voting twice.
00:02:33.860 --> 00:02:34.560
That's good.
00:02:34.560 --> 00:02:35.980
I like that.
00:02:35.980 --> 00:02:37.005
How about opposed?
00:02:39.930 --> 00:02:43.450
So that was really
close to a tie.
00:02:43.450 --> 00:02:44.780
All right.
00:02:44.780 --> 00:02:49.029
But I think the there was
slightly more opposed.
00:02:49.029 --> 00:02:49.570
I don't know.
00:02:49.570 --> 00:02:52.000
You guys who are in the
back maybe could tell.
00:02:52.000 --> 00:02:53.750
Anyway it was pretty close.
00:02:53.750 --> 00:02:54.250
All right.
00:02:54.250 --> 00:02:57.117
So now I'm going-- because
this is a real life thing,
00:02:57.117 --> 00:02:58.200
I'm going to try to do it.
00:02:58.200 --> 00:02:59.476
All right?
00:02:59.476 --> 00:03:00.550
All right.
00:03:00.550 --> 00:03:04.240
So now I'm going to tell
you what the trick is.
00:03:04.240 --> 00:03:08.890
The trick is to do it backwards.
00:03:08.890 --> 00:03:11.230
When most people are
playing with blocks,
00:03:11.230 --> 00:03:14.420
they decide to build
it from the bottom up.
00:03:14.420 --> 00:03:15.670
Right?
00:03:15.670 --> 00:03:19.280
But we're going to build
it from the top down,
00:03:19.280 --> 00:03:20.581
from the top down.
00:03:20.581 --> 00:03:22.080
And that's going
to make it possible
00:03:22.080 --> 00:03:25.080
for us to do the optimal
thing at each stage.
00:03:25.080 --> 00:03:28.400
So when I build it from the
top down, the best I can do
00:03:28.400 --> 00:03:30.520
is well, it'll fall off.
00:03:30.520 --> 00:03:34.040
I need to have it you
know, halfway across.
00:03:34.040 --> 00:03:35.470
That's the best I can do.
00:03:35.470 --> 00:03:38.000
So the top one I'm going
to build like that.
00:03:38.000 --> 00:03:41.730
I'm going to take it as
far to the left as I can.
00:03:41.730 --> 00:03:43.910
And then I'm going to
put the next one down
00:03:43.910 --> 00:03:46.160
as far to the left as I can.
00:03:46.160 --> 00:03:50.450
And then the next one as
far to the left as I can.
00:03:50.450 --> 00:03:51.740
That was a little too far.
00:03:51.740 --> 00:03:55.350
And then I'm going to do the
next one as far to the left
00:03:55.350 --> 00:03:56.382
as I can.
00:03:56.382 --> 00:03:58.250
And then I'm going
to do the next one --
00:03:58.250 --> 00:04:04.450
well let's line it up first --
as far to the left as I can.
00:04:04.450 --> 00:04:05.150
OK?
00:04:05.150 --> 00:04:09.036
And then the next one as
far to the left as I can.
00:04:09.036 --> 00:04:09.920
All right.
00:04:09.920 --> 00:04:15.700
Now those of you who are in
this line can see, all right,
00:04:15.700 --> 00:04:16.650
I succeeded.
00:04:16.650 --> 00:04:18.740
All right, that's over the edge.
00:04:18.740 --> 00:04:20.050
All right?
00:04:20.050 --> 00:04:21.604
So it can be done.
00:04:21.604 --> 00:04:22.390
All right.
00:04:25.450 --> 00:04:26.590
All right.
00:04:26.590 --> 00:04:32.750
So now we know that we can
get farther than you know,
00:04:32.750 --> 00:04:34.500
we can make it overflow.
00:04:34.500 --> 00:04:37.390
So the question now
is, how far can I get?
00:04:37.390 --> 00:04:38.240
OK.
00:04:38.240 --> 00:04:40.630
Do you think I can get to here?
00:04:40.630 --> 00:04:43.260
Can I get to the end over here?
00:04:43.260 --> 00:04:50.022
So how many people think I
can get this far over to here?
00:04:50.022 --> 00:04:51.730
How many people think
I can get this far?
00:04:51.730 --> 00:04:53.525
Well you know,
remember, I'm going
00:04:53.525 --> 00:04:55.519
to have to use more
than just this one
00:04:55.519 --> 00:04:56.560
more block that I've got.
00:04:56.560 --> 00:04:58.280
I don't, right?
00:04:58.280 --> 00:05:00.900
Obviously I'm
thinking, actually I do
00:05:00.900 --> 00:05:02.330
have some more blocks at home.
00:05:02.330 --> 00:05:03.294
But, OK.
00:05:03.294 --> 00:05:04.085
We're not going to.
00:05:04.085 --> 00:05:06.510
But anyway, do you think
I can get over to here?
00:05:06.510 --> 00:05:09.710
How many people say yes?
00:05:09.710 --> 00:05:13.060
And how many people say no?
00:05:13.060 --> 00:05:15.180
More people said no then yes.
00:05:15.180 --> 00:05:15.680
All right.
00:05:15.680 --> 00:05:18.480
So maybe the stopping place
is some mysterious number
00:05:18.480 --> 00:05:19.995
in between here.
00:05:19.995 --> 00:05:20.690
All right?
00:05:20.690 --> 00:05:21.720
Well OK.
00:05:21.720 --> 00:05:24.840
So now we're going
to do the arithmetic.
00:05:24.840 --> 00:05:28.870
And we're going to figure out
what happens with this problem.
00:05:28.870 --> 00:05:29.500
OK?
00:05:29.500 --> 00:05:32.175
So let's do it.
00:05:32.175 --> 00:05:35.760
All right, so now
again the idea is,
00:05:35.760 --> 00:05:46.710
the idea is we're going to start
with the top, the top block.
00:05:46.710 --> 00:05:48.450
We'll call that
block number one.
00:05:53.070 --> 00:05:56.295
And then the farthest,
if you like, to the right
00:05:56.295 --> 00:05:57.920
that you can put a
block underneath it,
00:05:57.920 --> 00:05:59.070
is exactly halfway.
00:06:01.800 --> 00:06:04.610
All right, well, that's
the best job I can do.
00:06:04.610 --> 00:06:08.270
Now in order to make my
units work out easily,
00:06:08.270 --> 00:06:13.160
I'm going to decide to call
the length of the block 2.
00:06:13.160 --> 00:06:14.090
All right?
00:06:14.090 --> 00:06:17.000
And that means if I
start at location 0,
00:06:17.000 --> 00:06:22.550
then the first place where I
am is supposed to be halfway.
00:06:22.550 --> 00:06:24.070
And that will be 1.
00:06:26.650 --> 00:06:31.660
OK so the first step in the
process is 1 more to the right.
00:06:31.660 --> 00:06:33.610
Or if you like, if I
were building up --
00:06:33.610 --> 00:06:36.068
which is what you would actually
have to do in real life --
00:06:36.068 --> 00:06:39.280
it would be 1 to the left.
00:06:39.280 --> 00:06:41.610
OK now the next one.
00:06:41.610 --> 00:06:44.294
Now here is the
way that you start
00:06:44.294 --> 00:06:45.460
figuring out the arithmetic.
00:06:45.460 --> 00:06:48.190
The next one is based
on a physical principle.
00:06:48.190 --> 00:06:53.350
Which is that the farthest
I can stick this next block
00:06:53.350 --> 00:06:57.950
underneath is what's called the
center of mass of these two,
00:06:57.950 --> 00:07:00.136
which is exactly halfway here.
00:07:00.136 --> 00:07:01.760
That is, there's a
quarter of this guy,
00:07:01.760 --> 00:07:04.400
and a quarter of that
guy balancing each other.
00:07:04.400 --> 00:07:04.900
Right?
00:07:04.900 --> 00:07:06.160
So that's as far as I can go.
00:07:06.160 --> 00:07:08.625
If I go farther than
that, it'll fall over.
00:07:08.625 --> 00:07:10.600
So that's the absolute
farthest I can do.
00:07:10.600 --> 00:07:14.630
So the next block is
going to be over here.
00:07:14.630 --> 00:07:18.210
And a quarter of 2 is 1/2.
00:07:18.210 --> 00:07:21.552
So this is 3/2 here.
00:07:21.552 --> 00:07:24.430
All right so we went to 1.
00:07:24.430 --> 00:07:27.960
We went to 3/2 here.
00:07:27.960 --> 00:07:31.300
And then I'm going to keep on
going with this eventually.
00:07:31.300 --> 00:07:33.140
All right so we're
going to figure out
00:07:33.140 --> 00:07:36.770
what happens with this stack.
00:07:36.770 --> 00:07:37.774
Question?
00:07:37.774 --> 00:07:40.065
AUDIENCE: How do
you know that this
00:07:40.065 --> 00:07:42.670
is the best way to optimize?
00:07:42.670 --> 00:07:44.515
PROFESSOR: The
question is how do I
00:07:44.515 --> 00:07:46.870
know that this is the
best way to optimize?
00:07:46.870 --> 00:07:49.099
I can't answer that question.
00:07:49.099 --> 00:07:50.640
But I can tell you
that it's the best
00:07:50.640 --> 00:07:52.735
way if I start with
a top like this,
00:07:52.735 --> 00:07:53.860
and the next one like this.
00:07:53.860 --> 00:07:56.100
Right, because I'm
doing the farthest
00:07:56.100 --> 00:07:57.470
possible at each stage.
00:07:57.470 --> 00:07:59.840
That actually has a name
in computer science, that's
00:07:59.840 --> 00:08:01.610
called the greedy algorithm.
00:08:01.610 --> 00:08:04.520
I'm trying to do the best
possible at each stage.
00:08:04.520 --> 00:08:07.040
The greedy algorithm
starting from the bottom
00:08:07.040 --> 00:08:09.420
is an extremely bad strategy.
00:08:09.420 --> 00:08:12.790
Because when you do that,
you stack it this way,
00:08:12.790 --> 00:08:14.120
and it almost falls over.
00:08:14.120 --> 00:08:16.050
And then the next time
you can't do anything.
00:08:16.050 --> 00:08:19.030
So the greedy algorithm is
terrible from the bottom.
00:08:19.030 --> 00:08:21.225
This is the greedy algorithm
starting from the top,
00:08:21.225 --> 00:08:23.455
and it turns out
to do much better
00:08:23.455 --> 00:08:25.580
then the greedy algorithm
starting from the bottom.
00:08:25.580 --> 00:08:27.913
But of course I'm not addressing
whether there might not
00:08:27.913 --> 00:08:31.500
be some other incredibly clever
strategy where I wiggle around
00:08:31.500 --> 00:08:34.260
and make it go up.
00:08:34.260 --> 00:08:35.676
I'm not addressing
that question.
00:08:35.676 --> 00:08:36.130
All right?
00:08:36.130 --> 00:08:37.838
It turns out this is
the best you can do.
00:08:37.838 --> 00:08:40.790
But that's not clear.
00:08:40.790 --> 00:08:44.650
All right so now, here
we have this thing.
00:08:44.650 --> 00:08:47.600
And now I have to figure out
what the arithmetic pattern is,
00:08:47.600 --> 00:08:50.340
so that I can figure out what
I was doing with those shapes.
00:08:53.850 --> 00:08:58.132
So let's figure out a
thought experiment here.
00:08:58.132 --> 00:08:59.526
All right?
00:08:59.526 --> 00:09:00.900
Now the thought
experiment I want
00:09:00.900 --> 00:09:03.450
to imagine for
you is, you've got
00:09:03.450 --> 00:09:10.070
a stack of a bunch of blocks,
and this is the first N blocks.
00:09:14.150 --> 00:09:15.310
All right?
00:09:15.310 --> 00:09:19.010
And now we're going to
put one underneath it.
00:09:19.010 --> 00:09:21.650
And what we're
going to figure out
00:09:21.650 --> 00:09:26.340
is the center of mass
of those N blocks,
00:09:26.340 --> 00:09:30.960
which I'm going to
call C sub N. OK.
00:09:30.960 --> 00:09:32.615
And that's the place
where I'm going
00:09:32.615 --> 00:09:34.320
to put this very next block.
00:09:34.320 --> 00:09:36.480
I'll put it in a
different color here.
00:09:36.480 --> 00:09:39.560
Here's the new--
the next block over.
00:09:39.560 --> 00:09:42.360
And the next block over
is the (N+1)st block.
00:09:49.900 --> 00:09:54.380
And now I want you to think
about what's going on here.
00:09:54.380 --> 00:09:56.880
If the center of mass
of the first N blocks
00:09:56.880 --> 00:10:01.700
is this number, this new
one, it's of length 2.
00:10:01.700 --> 00:10:04.470
And its center of
mass is 1 further
00:10:04.470 --> 00:10:07.850
to the right than the center
of mass that we had before.
00:10:07.850 --> 00:10:10.930
So in other words, I've added to
this configuration of N blocks
00:10:10.930 --> 00:10:13.220
one more block,
which is shifted.
00:10:13.220 --> 00:10:15.650
Whose center mass
is not lined up
00:10:15.650 --> 00:10:17.630
with the center of mass
of this, but actually
00:10:17.630 --> 00:10:20.200
over farther to the right.
00:10:20.200 --> 00:10:23.170
All right so the
new center of mass
00:10:23.170 --> 00:10:27.060
of this new block-- And this is
the extra piece of information
00:10:27.060 --> 00:10:29.670
that I want to observe,
is that this thing has
00:10:29.670 --> 00:10:36.740
a center of mass at C_N + 1.
00:10:36.740 --> 00:10:39.870
It's 1 unit over because
this total length is 2.
00:10:39.870 --> 00:10:45.561
So right in the middle there is
1 over, according to my units.
00:10:45.561 --> 00:10:48.240
All right now this is
going to make it possible
00:10:48.240 --> 00:10:51.970
for me to figure out what
the new center of mass is.
00:10:51.970 --> 00:11:04.860
So C_(N+1) is the center
of mass of N+1 blocks.
00:11:04.860 --> 00:11:08.910
Now this is really only in the
horizontal variable, right?
00:11:08.910 --> 00:11:11.260
I'm not keeping track of the
center of mass-- Actually
00:11:11.260 --> 00:11:13.510
this thing is hard to build
because the center of mass
00:11:13.510 --> 00:11:14.240
is also rising.
00:11:14.240 --> 00:11:15.830
It's getting higher and higher.
00:11:15.830 --> 00:11:19.510
But I'm only keeping track of
its left-right characteristic.
00:11:19.510 --> 00:11:21.590
So this is the
x-coordinate of it.
00:11:27.760 --> 00:11:31.590
All right so now
here's the idea.
00:11:31.590 --> 00:11:34.640
I'm combining the white
ones, the N blocks,
00:11:34.640 --> 00:11:37.630
with the pink one, which
is the one on the bottom.
00:11:37.630 --> 00:11:41.020
And there are N
of the white ones.
00:11:41.020 --> 00:11:42.520
And there's 1 of the pink one.
00:11:42.520 --> 00:11:44.811
And so in order to get the
center of mass of the whole,
00:11:44.811 --> 00:11:47.700
I have to take the weighted
average of the two.
00:11:47.700 --> 00:11:56.090
That's N*C_N plus 1 times the
center of mass of the pink one,
00:11:56.090 --> 00:11:59.665
which is C_N + 1.
00:11:59.665 --> 00:12:02.040
And then I have to divide --
if it's the weighted average
00:12:02.040 --> 00:12:06.560
of the total of N +
1 blocks -- by N + 1.
00:12:06.560 --> 00:12:09.090
This is going to give me
the new center of mass
00:12:09.090 --> 00:12:11.855
of my configuration
at the (N+1)st stage.
00:12:14.800 --> 00:12:16.330
And now I can just
do the arithmetic
00:12:16.330 --> 00:12:19.340
and figure out what this is.
00:12:19.340 --> 00:12:21.810
And the two C_Ns combine.
00:12:21.810 --> 00:12:27.560
I get (N+1)C_N +
1, divided by N+1.
00:12:30.250 --> 00:12:33.680
And if I combine these two
things and do the cancellation,
00:12:33.680 --> 00:12:36.140
that gives me this
recurrence formula,
00:12:36.140 --> 00:12:41.040
C_(N+1) is equal to C_N
plus-- There's a little extra.
00:12:41.040 --> 00:12:42.010
These two cancel.
00:12:42.010 --> 00:12:43.380
That gives me the C_N.
00:12:43.380 --> 00:12:46.740
But then I also have 1/(N+1).
00:12:54.230 --> 00:12:57.270
Well that's how much gain I
can get in the center of mass
00:12:57.270 --> 00:12:58.620
by adding one more block.
00:12:58.620 --> 00:13:01.520
That's how much I can
shift things over,
00:13:01.520 --> 00:13:03.550
depending on how we're
thinking of things,
00:13:03.550 --> 00:13:05.700
to the left or the
right, depending on which
00:13:05.700 --> 00:13:07.137
direction we're building them.
00:13:10.960 --> 00:13:13.840
All right, so now I'm going
to work out the formulas.
00:13:13.840 --> 00:13:17.680
First of all C_1, that was
the center of the first block.
00:13:17.680 --> 00:13:20.490
I put its left end at 0; the
center of the first block
00:13:20.490 --> 00:13:22.250
is at 1.
00:13:22.250 --> 00:13:25.910
That means that C_1 is 1.
00:13:25.910 --> 00:13:27.310
OK?
00:13:27.310 --> 00:13:31.090
C_2 according to this
formula-- And actually I've
00:13:31.090 --> 00:13:35.530
worked it out, we'll check
it in a-- C_2 is C_1 + 1/2.
00:13:35.530 --> 00:13:37.380
All right, so that's
the case N = 1.
00:13:37.380 --> 00:13:40.130
So this is 1 + 1/2.
00:13:40.130 --> 00:13:43.420
That's what we already did.
00:13:43.420 --> 00:13:45.790
That's the 3/2 number.
00:13:45.790 --> 00:13:51.320
Now the next one is C_2 + 1/3.
00:13:51.320 --> 00:13:53.530
That's the formula again.
00:13:53.530 --> 00:13:59.590
And so that comes out
to be 1 + 1/2 + 1/3.
00:13:59.590 --> 00:14:03.100
And now you can see
what the pattern is.
00:14:03.100 --> 00:14:06.810
C_N-- If you just
keep on going here,
00:14:06.810 --> 00:14:14.130
C_N is going to be 1
+ 1/2 + 1/3 + 1/4...
00:14:14.130 --> 00:14:21.410
plus 1/N.
00:14:21.410 --> 00:14:25.600
So now I would like
you to vote again.
00:14:25.600 --> 00:14:27.790
Do you think I can-- Now
that we have the formula,
00:14:27.790 --> 00:14:30.830
do you think I can
get over to here?
00:14:30.830 --> 00:14:33.180
How many people think
I can get over to here?
00:14:36.160 --> 00:14:40.410
How many people think I
can't get over to here?
00:14:40.410 --> 00:14:42.700
There's still a lot
of people who do.
00:14:42.700 --> 00:14:46.420
So it's still almost 50/50.
00:14:46.420 --> 00:14:47.470
That's amazing.
00:14:47.470 --> 00:14:49.800
Well so we'll address
that in a few minutes.
00:14:49.800 --> 00:14:52.690
So now let me tell
you what's going on.
00:14:52.690 --> 00:14:56.050
This C_N of course, is the same
as what we called last time
00:14:56.050 --> 00:14:57.530
S_N.
00:14:57.530 --> 00:15:01.230
And remember that we actually
estimated the size of this guy.
00:15:01.230 --> 00:15:04.620
This is related to what's
called the harmonic series.
00:15:04.620 --> 00:15:08.120
And what we showed
was that log N
00:15:08.120 --> 00:15:14.690
is less than S_N, which
is less than S_N + 1.
00:15:14.690 --> 00:15:15.335
All right?
00:15:17.910 --> 00:15:20.360
Now I'm going to
call your attention
00:15:20.360 --> 00:15:24.260
to the red part, which
is the divergence
00:15:24.260 --> 00:15:28.500
part of this estimate, which
is this one for the time being,
00:15:28.500 --> 00:15:30.290
all right.
00:15:30.290 --> 00:15:32.780
Just saying that this
thing is growing.
00:15:32.780 --> 00:15:41.270
And what this is saying is
that as N goes to infinity,
00:15:41.270 --> 00:15:48.810
log N goes to
infinity, So that means
00:15:48.810 --> 00:15:57.220
that S_N goes to infinity,
because of this inequality
00:15:57.220 --> 00:15:57.980
here.
00:15:57.980 --> 00:16:01.780
It's bigger than log N.
And so if N is big enough,
00:16:01.780 --> 00:16:04.002
we can get as far as we like.
00:16:04.002 --> 00:16:06.030
All right?
00:16:06.030 --> 00:16:08.312
So I can get to here.
00:16:08.312 --> 00:16:10.270
And at least half of you,
at least the ones who
00:16:10.270 --> 00:16:12.224
voted, that was-- I don't know.
00:16:12.224 --> 00:16:13.890
We have a quorum here,
but I'm not sure.
00:16:13.890 --> 00:16:16.470
We certainly didn't have
a majority on either side.
00:16:16.470 --> 00:16:19.420
Anyway this thing
does go to infinity.
00:16:19.420 --> 00:16:21.330
So in principle, if
I had enough blocks,
00:16:21.330 --> 00:16:25.216
I could get it over to here.
00:16:25.216 --> 00:16:26.590
All right, and
that's the meaning
00:16:26.590 --> 00:16:28.300
of divergence in this case.
00:16:32.430 --> 00:16:36.670
On the other hand, I want
to discuss with you--
00:16:36.670 --> 00:16:38.600
And the reason why
I use this example,
00:16:38.600 --> 00:16:40.390
is I want to discuss
with you also what's
00:16:40.390 --> 00:16:45.460
going on with this
other inequality here,
00:16:45.460 --> 00:16:49.360
and what its significance is.
00:16:49.360 --> 00:16:52.970
Which is that it's going to
take us a lot of numbers N,
00:16:52.970 --> 00:16:57.050
a lot of blocks, to get
up to a certain level.
00:16:57.050 --> 00:17:00.180
In other words, I can't do it
with just eight blocks or nine
00:17:00.180 --> 00:17:00.800
blocks.
00:17:00.800 --> 00:17:02.258
In order to get
over here, I'd have
00:17:02.258 --> 00:17:06.000
to use quite a few of them.
00:17:06.000 --> 00:17:11.120
So let's just see
how many it is.
00:17:11.120 --> 00:17:14.230
So I worked this out carefully.
00:17:14.230 --> 00:17:15.645
And let's see what I got.
00:17:18.150 --> 00:17:31.760
So to get across the
lab tables, all right.
00:17:31.760 --> 00:17:35.780
This distance here, I
already did this secretly.
00:17:35.780 --> 00:17:38.985
But I don't actually even have
enough of these to show you.
00:17:38.985 --> 00:17:43.834
But, well 1, 2, 3,
4, 5, 6, and 1/2.
00:17:43.834 --> 00:17:44.750
I guess that's enough.
00:17:44.750 --> 00:17:46.240
So it's 6 and a half.
00:17:46.240 --> 00:17:50.000
So it's two lab tables
is 13 of these blocks.
00:17:50.000 --> 00:17:51.190
All right.
00:17:51.190 --> 00:18:00.310
So there are 13 blocks,
which is equal to 26 units.
00:18:00.310 --> 00:18:04.020
OK, that's how far
to get across I need.
00:18:04.020 --> 00:18:06.090
And the first one is already 2.
00:18:06.090 --> 00:18:09.470
So it's really 26
minus 2, which is 24.
00:18:09.470 --> 00:18:11.450
Which that's what I need.
00:18:11.450 --> 00:18:13.110
OK.
00:18:13.110 --> 00:18:22.910
So I need log N to be equal
to 24, roughly speaking,
00:18:22.910 --> 00:18:25.090
in order to get that far.
00:18:25.090 --> 00:18:28.195
So let's just see
how big that is.
00:18:28.195 --> 00:18:30.300
All right.
00:18:30.300 --> 00:18:31.760
I think I worked this out.
00:18:42.010 --> 00:18:43.170
So let's see.
00:18:43.170 --> 00:18:48.020
That means that N
is equal to e^24--
00:18:48.020 --> 00:18:57.160
and if you realize that these
blocks are 3 centimeters high--
00:18:57.160 --> 00:19:00.450
OK let's see how many
that we would need here.
00:19:00.450 --> 00:19:02.210
That's kind of a lot.
00:19:02.210 --> 00:19:09.440
Let's see, it's 3
centimeters times e^24,
00:19:09.440 --> 00:19:15.660
which is about 8*10^8 meters.
00:19:15.660 --> 00:19:17.600
OK.
00:19:17.600 --> 00:19:20.810
And that is twice the
distance to the moon.
00:19:32.310 --> 00:19:35.190
So OK, so I could do it maybe.
00:19:35.190 --> 00:19:37.936
But I would need
a lot of blocks.
00:19:37.936 --> 00:19:38.435
Right?
00:19:38.435 --> 00:19:42.190
So that's not very
plausible here, all right.
00:19:42.190 --> 00:19:44.380
So those of you who
voted against this
00:19:44.380 --> 00:19:47.080
were actually sort
of half right.
00:19:47.080 --> 00:19:48.630
And in fact, if you
wanted to get it
00:19:48.630 --> 00:19:52.300
to the wall over there,
which is over 30 feet,
00:19:52.300 --> 00:19:55.330
the height would be
about the diameter
00:19:55.330 --> 00:19:57.360
of the observable universe.
00:19:57.360 --> 00:20:01.580
That's kind of a long way.
00:20:01.580 --> 00:20:05.780
There's one other thing
that I wanted to point out
00:20:05.780 --> 00:20:09.340
to you about this shape here.
00:20:09.340 --> 00:20:13.516
Which is that if you
lean to the left, right,
00:20:13.516 --> 00:20:14.890
if you put your
head like this --
00:20:14.890 --> 00:20:17.510
of course you have to be on
your side to look at it --
00:20:17.510 --> 00:20:25.810
this curve is the shape
of a logarithmic curve.
00:20:25.810 --> 00:20:29.240
So in other words, if you think
of the vertical as the x-axis,
00:20:29.240 --> 00:20:32.010
and the horizontal that
way is the vertical,
00:20:32.010 --> 00:20:34.610
is the up direction,
then this thing
00:20:34.610 --> 00:20:38.510
is growing very, very,
very, very slowly.
00:20:38.510 --> 00:20:43.030
If you send the x-axis all
the way up to the moon,
00:20:43.030 --> 00:20:47.540
the graph still hasn't gotten
across the lab tables here.
00:20:47.540 --> 00:20:48.905
It's only partway there.
00:20:48.905 --> 00:20:52.110
If you go twice the distance
to the moon up that way,
00:20:52.110 --> 00:20:54.340
it's gotten finally to that end.
00:20:54.340 --> 00:20:56.990
All right so that's how
slowly the logarithm grows.
00:20:56.990 --> 00:20:58.189
It grows very, very slowly.
00:20:58.189 --> 00:21:00.730
And if you look at it another
way, if you stand on your head,
00:21:00.730 --> 00:21:05.420
you can see an
exponential curve.
00:21:05.420 --> 00:21:08.470
So you get some sense as
to the growth properties
00:21:08.470 --> 00:21:10.420
of these functions.
00:21:10.420 --> 00:21:14.620
And fortunately these
are protecting us
00:21:14.620 --> 00:21:18.550
from all kinds of
stuff that would
00:21:18.550 --> 00:21:20.200
happen if there
weren't exponentially
00:21:20.200 --> 00:21:21.750
small tails in the world.
00:21:21.750 --> 00:21:23.670
Like you know, I could
walk through this wall
00:21:23.670 --> 00:21:27.000
which I wouldn't like doing.
00:21:27.000 --> 00:21:32.410
OK, now so this is
our last example.
00:21:32.410 --> 00:21:35.000
And the important
number, unfortunately we
00:21:35.000 --> 00:21:36.790
didn't discover another
important number.
00:21:36.790 --> 00:21:40.250
There wasn't an amazing number
place where this stopped.
00:21:40.250 --> 00:21:44.160
All we discovered again is
some property of infinity.
00:21:44.160 --> 00:21:46.150
So infinity is
still a nice number.
00:21:46.150 --> 00:21:50.940
And the theme here is just that
infinity isn't just one thing,
00:21:50.940 --> 00:21:54.009
it has a character which
is a rate of growth.
00:21:54.009 --> 00:21:55.550
And you shouldn't
just think of there
00:21:55.550 --> 00:21:57.072
being one order of infinity.
00:21:57.072 --> 00:21:58.530
There are lots of
different orders.
00:21:58.530 --> 00:22:01.280
And some of them have
different meaning from others.
00:22:01.280 --> 00:22:03.880
All right so that's
the theme I wanted
00:22:03.880 --> 00:22:07.900
to do, and just have a
visceral example of infinity.
00:22:07.900 --> 00:22:13.570
Now, we're going to move
on now to some other kinds
00:22:13.570 --> 00:22:16.210
of techniques.
00:22:16.210 --> 00:22:20.790
And this is going to
be our last subject.
00:22:20.790 --> 00:22:23.820
What we're going
to talk about is
00:22:23.820 --> 00:22:27.320
what are known as power series.
00:22:27.320 --> 00:22:29.920
And we've already seen
our first power series.
00:22:32.710 --> 00:22:35.240
And I'm going to
remind you of that.
00:22:43.980 --> 00:22:45.490
Here we are with power series.
00:22:51.490 --> 00:22:53.980
Our first series was this one.
00:22:58.870 --> 00:23:05.200
And we mentioned last time
that it was equal to 1/(1-x),
00:23:05.200 --> 00:23:06.430
for x less than 1.
00:23:09.770 --> 00:23:12.040
Well this one is known
as the geometric series.
00:23:12.040 --> 00:23:15.050
You didn't use the letter x
last time, I used the letter a.
00:23:15.050 --> 00:23:16.910
But this is known as
the geometric series.
00:23:24.950 --> 00:23:31.550
Now I'm going to show you
one reason why this is true,
00:23:31.550 --> 00:23:33.900
why the formula holds.
00:23:33.900 --> 00:23:36.050
And it's just the
kind of manipulation
00:23:36.050 --> 00:23:41.270
that was done when these
things were first introduced.
00:23:41.270 --> 00:23:46.250
And here's the idea of a proof.
00:23:46.250 --> 00:23:52.670
So suppose that this sum
is equal to some number
00:23:52.670 --> 00:23:57.360
S, which is the sum of
all of these numbers here.
00:24:00.540 --> 00:24:02.130
The first thing
that I'm going to do
00:24:02.130 --> 00:24:05.500
is I'm going to multiply by x.
00:24:05.500 --> 00:24:08.326
OK, so if I multiply by x.
00:24:08.326 --> 00:24:09.560
Let's think about that.
00:24:09.560 --> 00:24:13.810
I multiply by x on both the
left and the right-hand side.
00:24:13.810 --> 00:24:20.990
Then on the left side, I get x
+ x^2 + x^3 plus, and so forth.
00:24:20.990 --> 00:24:22.810
And on the right
side, I get S x.
00:24:27.380 --> 00:24:29.990
And now I'm going
to subtract the two
00:24:29.990 --> 00:24:32.910
equations, one from the other.
00:24:32.910 --> 00:24:36.230
And there's a very, very
substantial cancellation.
00:24:36.230 --> 00:24:39.180
This whole tail here
gets canceled off.
00:24:39.180 --> 00:24:41.030
And the only thing
that's left is the 1.
00:24:41.030 --> 00:24:45.950
So when I subtract, I get
1 on the left-hand side.
00:24:45.950 --> 00:24:53.000
And on the right-hand
side, I get S - S x.
00:24:53.000 --> 00:24:53.500
All right?
00:24:58.900 --> 00:25:03.850
And now that can be
rewritten as S(1-x).
00:25:03.850 --> 00:25:06.180
And so I've got my formula here.
00:25:06.180 --> 00:25:13.553
This is 1/(1-x) = S. All right.
00:25:17.850 --> 00:25:26.340
Now this reasoning has one flaw.
00:25:26.340 --> 00:25:28.120
It's not complete.
00:25:28.120 --> 00:25:33.080
And this reasoning
is basically correct.
00:25:33.080 --> 00:25:45.680
But it's incomplete because
it requires that S exists.
00:25:50.270 --> 00:25:55.100
For example, it doesn't make
any sense in the case x = 1.
00:25:55.100 --> 00:25:57.950
So for example in
the case x = 1,
00:25:57.950 --> 00:26:01.710
we have 1 + 1 +
1 plus et cetera,
00:26:01.710 --> 00:26:04.830
equals whatever we call S. And
then when we multiply through
00:26:04.830 --> 00:26:08.798
by 1, we get 1 + 1 + 1 plus...
00:26:08.798 --> 00:26:10.640
equals S*1.
00:26:10.640 --> 00:26:12.560
And now you see that
the subtraction gives us
00:26:12.560 --> 00:26:16.330
infinity minus infinity is equal
to infinity minus infinity.
00:26:16.330 --> 00:26:19.790
That's what's really going on
in the argument in this context.
00:26:19.790 --> 00:26:21.340
So it's just nonsense.
00:26:21.340 --> 00:26:24.380
I mean it doesn't give
us anything meaningful.
00:26:24.380 --> 00:26:27.090
So this argument, it's great.
00:26:27.090 --> 00:26:30.000
And it gives us the right
answer, but not always.
00:26:30.000 --> 00:26:33.130
And the times when it gives us
the answer, the correct answer,
00:26:33.130 --> 00:26:36.800
is when the series
is convergent.
00:26:36.800 --> 00:26:38.800
And that's why we care
about convergence.
00:26:38.800 --> 00:26:42.060
Because we want manipulations
like this to be allowed.
00:26:47.780 --> 00:26:50.200
So the good case,
this is the red case
00:26:50.200 --> 00:26:52.730
that we were
describing last time.
00:26:52.730 --> 00:26:54.790
That's the bad case.
00:26:54.790 --> 00:26:58.820
But what we want is the good
case, the convergent case.
00:26:58.820 --> 00:27:01.770
And that is the case
when x is less than 1.
00:27:01.770 --> 00:27:03.182
So this is the convergent case.
00:27:11.290 --> 00:27:11.960
Yep.
00:27:11.960 --> 00:27:14.140
OK, so they're much
more detailed things
00:27:14.140 --> 00:27:15.710
to check exactly
what's going on.
00:27:15.710 --> 00:27:18.090
But I'm going to just say
general words about how
00:27:18.090 --> 00:27:20.099
you recognize convergence.
00:27:20.099 --> 00:27:22.140
And then we're not going
to worry about-- so much
00:27:22.140 --> 00:27:24.740
about convergence, because
it works very, very well.
00:27:24.740 --> 00:27:27.480
And it's always easy
to diagnose when
00:27:27.480 --> 00:27:31.590
there's convergence
with a power series.
00:27:31.590 --> 00:27:33.450
All right so here's
the general setup.
00:27:44.700 --> 00:27:48.510
The general setup
is that we have
00:27:48.510 --> 00:27:55.600
not just the coefficients 1 all
the time, but any numbers here,
00:27:55.600 --> 00:27:56.660
dot, dot, dot.
00:27:56.660 --> 00:27:59.310
And we abbreviate that with
the summation notation.
00:27:59.310 --> 00:28:05.360
This is the sum a_n x^n,
n equals 0 to infinity.
00:28:05.360 --> 00:28:07.260
And that's what's known
as a power series.
00:28:12.370 --> 00:28:16.370
Fortunately there is
a very simple rule
00:28:16.370 --> 00:28:20.260
about how power series converge.
00:28:20.260 --> 00:28:23.200
And it's the following.
00:28:23.200 --> 00:28:28.480
There's a magic number R which
depends on these numbers here
00:28:28.480 --> 00:28:30.160
such that-- And
this thing is known
00:28:30.160 --> 00:28:31.460
as a radius of convergence.
00:28:37.020 --> 00:28:40.090
In the problem that we had,
it's this number 1 here.
00:28:40.090 --> 00:28:43.030
This thing works
for x less than 1.
00:28:43.030 --> 00:28:46.190
In our case, it's
maybe x less than R. So
00:28:46.190 --> 00:28:47.960
that's some symmetric
interval, right?
00:28:47.960 --> 00:28:53.630
That's the same as minus
R less than x less than R,
00:28:53.630 --> 00:28:58.310
and so where
there's convergence.
00:28:58.310 --> 00:29:00.310
OK, where the series converges.
00:29:00.310 --> 00:29:00.810
Converges.
00:29:07.210 --> 00:29:12.150
And then there's
the region where
00:29:12.150 --> 00:29:16.500
every computation that you
give will give you nonsense.
00:29:16.500 --> 00:29:24.740
So x greater than R is
the sum a_n x^n diverges.
00:29:28.710 --> 00:29:42.030
And x equals R is very
delicate, borderline,
00:29:42.030 --> 00:29:44.990
and will not be used by us.
00:29:50.060 --> 00:29:54.750
OK, we're going to stick inside
the radius of convergence.
00:29:54.750 --> 00:29:57.620
Now the way you'll be
able to recognize this,
00:29:57.620 --> 00:29:58.920
is the following.
00:29:58.920 --> 00:30:03.110
What always happens
is that these numbers
00:30:03.110 --> 00:30:15.370
tend to 0 exponentially
fast, fast for x in R,
00:30:15.370 --> 00:30:26.600
and doesn't even tend to 0
at all for x greater than R.
00:30:26.600 --> 00:30:30.200
All right so it'll
be totally obvious.
00:30:30.200 --> 00:30:32.210
When you look at
this series here,
00:30:32.210 --> 00:30:34.040
what's happening
when x less than R
00:30:34.040 --> 00:30:37.400
is that the numbers are getting
smaller and smaller, less
00:30:37.400 --> 00:30:37.990
than 1.
00:30:37.990 --> 00:30:39.290
When x is bigger
than 1, the numbers
00:30:39.290 --> 00:30:40.539
are getting bigger and bigger.
00:30:40.539 --> 00:30:42.850
There's no chance that
the series converges.
00:30:42.850 --> 00:30:45.870
So that's going to be the
case with all power series.
00:30:45.870 --> 00:30:47.680
There's going to be a cutoff.
00:30:47.680 --> 00:30:49.369
And it'll be one
particular number.
00:30:49.369 --> 00:30:51.785
And below that it'll be obvious
that you have convergence,
00:30:51.785 --> 00:30:53.368
and you'll be able
to do computations.
00:30:53.368 --> 00:30:55.690
And above that every
formula will be wrong
00:30:55.690 --> 00:30:57.220
and won't make sense.
00:30:57.220 --> 00:30:59.370
So it's a very clean thing.
00:30:59.370 --> 00:31:01.660
There is this very
subtle borderline,
00:31:01.660 --> 00:31:04.350
but we're not going to
discuss that in this class.
00:31:04.350 --> 00:31:10.383
And it's actually not used in
direct studies of power series.
00:31:10.383 --> 00:31:13.820
AUDIENCE: How can you tell
when the numbers are declining
00:31:13.820 --> 00:31:16.780
exponentially fast, whereas
just-- In other words
00:31:16.780 --> 00:31:18.030
1/x [INAUDIBLE]?
00:31:18.030 --> 00:31:20.470
PROFESSOR: OK so,
the question is
00:31:20.470 --> 00:31:23.070
why was I able to tell
you this word here?
00:31:23.070 --> 00:31:25.510
Why was I able to tell you
not only is it going to 0,
00:31:25.510 --> 00:31:27.550
but it's going
exponentially fast?
00:31:27.550 --> 00:31:29.000
I'm telling you
extra information.
00:31:29.000 --> 00:31:32.380
I'm telling you it always
goes exponentially fast.
00:31:32.380 --> 00:31:33.556
You can identify it.
00:31:36.150 --> 00:31:37.620
In other words, you'll see it.
00:31:37.620 --> 00:31:39.490
And it will happen
every single time.
00:31:39.490 --> 00:31:41.600
I'm just promising you
that it works that way.
00:31:41.600 --> 00:31:43.900
And it's really
for the same reason
00:31:43.900 --> 00:31:46.840
that it works that way
here, that these are powers.
00:31:46.840 --> 00:31:48.760
And what's going on
over here is there are,
00:31:48.760 --> 00:31:52.181
it's close to powers,
with these a_n's.
00:31:52.181 --> 00:31:54.640
All right?
00:31:54.640 --> 00:31:56.960
There's a long discussion
of radius of convergence
00:31:56.960 --> 00:31:58.350
in many textbooks.
00:31:58.350 --> 00:32:04.670
But really it's not necessary,
all right, for this purpose?
00:32:04.670 --> 00:32:05.418
Yeah?
00:32:05.418 --> 00:32:07.090
AUDIENCE: How do you find R?
00:32:07.090 --> 00:32:08.400
PROFESSOR: The question
was how do you find R?
00:32:08.400 --> 00:32:10.316
Yes, so I just said,
there's a long discussion
00:32:10.316 --> 00:32:12.660
for how you find the radius
of convergence in textbooks.
00:32:12.660 --> 00:32:15.800
But we will not be
discussing that here.
00:32:15.800 --> 00:32:17.680
And it won't be
necessary for you.
00:32:17.680 --> 00:32:20.890
Because it will be obvious in
any given series what the R is.
00:32:20.890 --> 00:32:23.650
It will always
either 1 or infinity.
00:32:23.650 --> 00:32:25.850
It will always work for
all x, or maybe it'll
00:32:25.850 --> 00:32:26.820
stop at some point.
00:32:26.820 --> 00:32:29.280
But it'll be very
clear where it stops,
00:32:29.280 --> 00:32:33.201
as it is for the
geometric series.
00:32:33.201 --> 00:32:36.070
All right?
00:32:36.070 --> 00:32:39.639
OK, so now I need to
give you the basic facts,
00:32:39.639 --> 00:32:40.805
and give you a few examples.
00:32:44.400 --> 00:32:46.420
So why are we looking
at these series?
00:32:51.240 --> 00:32:55.010
Well the answer is we're
looking at these series
00:32:55.010 --> 00:32:59.270
because the role that
they play is exactly
00:32:59.270 --> 00:33:03.420
the reverse of
this equation here.
00:33:03.420 --> 00:33:06.540
That is -- and this is a theme
which I have tried to emphasize
00:33:06.540 --> 00:33:09.730
throughout this course -- you
can read equalities in two
00:33:09.730 --> 00:33:11.870
directions.
00:33:11.870 --> 00:33:15.610
Both are interesting, typically.
00:33:15.610 --> 00:33:18.116
You can think, I don't know
what the value of this is.
00:33:18.116 --> 00:33:19.240
Here's a way of evaluating.
00:33:19.240 --> 00:33:21.540
And in other words,
the right side
00:33:21.540 --> 00:33:23.090
is a formula for the left side.
00:33:23.090 --> 00:33:25.400
Or you can think
of the left side
00:33:25.400 --> 00:33:27.140
as being a formula
for the right side.
00:33:30.720 --> 00:33:34.230
And the idea of series is
that they're flexible enough
00:33:34.230 --> 00:33:35.780
to represent all
of the functions
00:33:35.780 --> 00:33:39.300
that we've encountered
in this course.
00:33:39.300 --> 00:33:42.069
This is the tool which is very
much like the decimal expansion
00:33:42.069 --> 00:33:43.610
which allows you to
represent numbers
00:33:43.610 --> 00:33:45.130
like the square root of 2.
00:33:45.130 --> 00:33:47.520
Now we're going to be
representing all the numbers,
00:33:47.520 --> 00:33:52.550
all the functions that we know:
e^x, arctangent, sine, cosine.
00:33:52.550 --> 00:33:55.280
All of those functions
become completely flexible,
00:33:55.280 --> 00:33:57.970
and completely available to us,
and computationally available
00:33:57.970 --> 00:33:59.900
to us directly.
00:33:59.900 --> 00:34:01.440
So that's what
this is a tool for.
00:34:01.440 --> 00:34:03.580
And it's just like
decimal expansions
00:34:03.580 --> 00:34:05.240
giving you handle
on all real numbers.
00:34:08.920 --> 00:34:12.110
So here's how it works.
00:34:12.110 --> 00:34:28.170
The rules for
convergent power series
00:34:28.170 --> 00:34:33.920
are just like polynomials.
00:34:41.280 --> 00:34:43.660
All of the manipulations
that you do for power series
00:34:43.660 --> 00:34:46.120
are essentially the
same as for polynomials.
00:34:46.120 --> 00:34:49.190
So what kinds of things
do we do with polynomials?
00:34:49.190 --> 00:34:50.130
We add them.
00:34:53.280 --> 00:34:54.445
We multiply them together.
00:34:57.630 --> 00:35:00.790
We do substitutions.
00:35:00.790 --> 00:35:01.290
Right?
00:35:01.290 --> 00:35:04.970
We take one function
of another function.
00:35:04.970 --> 00:35:06.030
We divide them.
00:35:10.110 --> 00:35:11.590
OK.
00:35:11.590 --> 00:35:15.040
And these are all really not
very surprising operations.
00:35:15.040 --> 00:35:17.710
And we will be able to do
them with power series too.
00:35:17.710 --> 00:35:20.370
The ones that are interesting,
really interesting
00:35:20.370 --> 00:35:24.360
for calculus, are the last two.
00:35:24.360 --> 00:35:29.790
We differentiate them,
and we integrate them.
00:35:34.070 --> 00:35:36.060
And all of these
operations we'll
00:35:36.060 --> 00:35:38.110
be able to do for
power series as well.
00:35:42.950 --> 00:35:49.120
So now let's explain
the high points of this.
00:35:49.120 --> 00:35:51.530
Which is mainly just
the differentiation
00:35:51.530 --> 00:35:53.110
and the integration part.
00:35:53.110 --> 00:36:03.200
So if I take a series
like this and so forth,
00:36:03.200 --> 00:36:08.054
the formula for its derivative
is just like polynomials.
00:36:08.054 --> 00:36:10.220
That's what I just said,
it's just like polynomials.
00:36:10.220 --> 00:36:12.570
So the derivative of
the constant is 0.
00:36:12.570 --> 00:36:15.850
The derivative of
this term is a_1.
00:36:15.850 --> 00:36:19.340
This one is plus 2 a_2 x 2 x.
00:36:19.340 --> 00:36:23.920
This one is 3 a_3
x^2, et cetera.
00:36:23.920 --> 00:36:27.070
That's the formula.
00:36:27.070 --> 00:36:40.672
Similarly if I
integrate, well there's
00:36:40.672 --> 00:36:42.130
an unknown constant
which I'm going
00:36:42.130 --> 00:36:44.930
to put first rather than last.
00:36:44.930 --> 00:36:46.927
Which corresponds sort
of to the a_0 term which
00:36:46.927 --> 00:36:48.010
is going to get wiped out.
00:36:48.010 --> 00:36:50.770
That a_0 term suddenly
becomes a_0 x.
00:36:50.770 --> 00:36:56.980
And the anti-derivative of
this next term is a_1 x^2 / 2.
00:36:56.980 --> 00:37:02.680
And the next term is a_2
x^3 / 3, and so forth.
00:37:05.200 --> 00:37:06.456
Yeah, question?
00:37:06.456 --> 00:37:08.357
AUDIENCE: Is that a
series or a polynomial?
00:37:08.357 --> 00:37:10.190
PROFESSOR: Is this a
series or a polynomial?
00:37:10.190 --> 00:37:11.180
Good question.
00:37:11.180 --> 00:37:14.700
It's a polynomial if it ends.
00:37:14.700 --> 00:37:19.490
If it goes on infinitely
far, then it's a series.
00:37:19.490 --> 00:37:22.420
They look practically the
same, polynomials and series.
00:37:22.420 --> 00:37:26.286
There's this little
dot, dot, dot here.
00:37:26.286 --> 00:37:27.660
Is this a series
or a polynomial?
00:37:27.660 --> 00:37:28.780
It's the same rule.
00:37:28.780 --> 00:37:30.460
If it stops at a
finite stage, this one
00:37:30.460 --> 00:37:32.210
stops at a finite stage.
00:37:32.210 --> 00:37:35.074
If it goes on forever,
it goes on forever.
00:37:35.074 --> 00:37:37.980
AUDIENCE: So I thought that the
series add up finite numbers.
00:37:37.980 --> 00:37:41.735
You can add up terms
of x in series?
00:37:45.520 --> 00:37:47.380
PROFESSOR: So an
interesting question.
00:37:47.380 --> 00:37:49.435
So the question
that was just asked
00:37:49.435 --> 00:37:53.060
is I thought that a series
added up finite numbers.
00:37:53.060 --> 00:37:55.140
You could add up x?
00:37:55.140 --> 00:37:56.900
That was what you said, right?
00:37:56.900 --> 00:38:01.140
OK now notice that
I pulled that off
00:38:01.140 --> 00:38:03.720
on you by changing the
letter a to the letter
00:38:03.720 --> 00:38:10.170
x at the very beginning
of this commentary here.
00:38:10.170 --> 00:38:11.760
This is a series.
00:38:11.760 --> 00:38:15.630
For each individual value
of x, it's a number.
00:38:15.630 --> 00:38:17.790
So in other words, if
I plug in here x = 1/2,
00:38:17.790 --> 00:38:20.650
I'm going to add 1
+ 1/2 + 1/4 + 1/8,
00:38:20.650 --> 00:38:22.470
and I'll get a
number which is 2.
00:38:22.470 --> 00:38:25.030
And I'll plug in a number over
here, and I'll get a number.
00:38:25.030 --> 00:38:27.820
On the other hand, I can do
this for each value of x.
00:38:27.820 --> 00:38:31.770
So the interpretation of this
is that it's a function of x.
00:38:31.770 --> 00:38:34.380
And similarly this
is a function of x.
00:38:34.380 --> 00:38:37.400
It works when you plug
in the possible values
00:38:37.400 --> 00:38:42.330
x between -1 and 1.
00:38:42.330 --> 00:38:44.760
So there's really no
distinction there, it's
00:38:44.760 --> 00:38:47.010
just I slipped it passed you.
00:38:47.010 --> 00:38:48.130
These are functions of x.
00:38:50.790 --> 00:38:53.090
And the notion of a
power series is this idea
00:38:53.090 --> 00:38:54.860
that you put
coefficients on a series,
00:38:54.860 --> 00:38:57.040
but then you allow
yourself the flexibility
00:38:57.040 --> 00:38:58.990
to stick powers here.
00:38:58.990 --> 00:39:01.884
And that's exactly
what we're doing.
00:39:01.884 --> 00:39:03.300
OK there are other
kinds of series
00:39:03.300 --> 00:39:05.210
where you stick other
interesting functions in here
00:39:05.210 --> 00:39:06.168
like sines and cosines.
00:39:06.168 --> 00:39:08.620
There are lots of other
series that people study.
00:39:08.620 --> 00:39:10.465
And these are the simplest ones.
00:39:10.465 --> 00:39:12.360
And all those
examples are extremely
00:39:12.360 --> 00:39:14.515
helpful for
representing functions.
00:39:14.515 --> 00:39:18.920
But we're only going to
do this example here.
00:39:18.920 --> 00:39:23.610
All right, so here
are the two rules.
00:39:23.610 --> 00:39:29.880
And now there's only one
other complication here
00:39:29.880 --> 00:39:35.350
which I have to explain
to you before giving you
00:39:35.350 --> 00:39:41.270
a bunch of examples to show you
that this works extremely well.
00:39:41.270 --> 00:39:43.960
And the last thing
that I have to do
00:39:43.960 --> 00:39:45.460
for you is explain
to you something
00:39:45.460 --> 00:39:46.580
called Taylor's formula.
00:39:55.280 --> 00:40:00.730
Taylor's formula is the way you
get from the representations
00:40:00.730 --> 00:40:04.430
that we're used to of functions,
to a representation in the form
00:40:04.430 --> 00:40:06.180
of these coefficients.
00:40:06.180 --> 00:40:08.240
When I gave you
the function e^x,
00:40:08.240 --> 00:40:11.280
it didn't look
like a polynomial.
00:40:11.280 --> 00:40:14.010
And we have to figure
out which of these guys
00:40:14.010 --> 00:40:19.090
it is, if it's going to
fall into our category here.
00:40:19.090 --> 00:40:20.370
And here's the formula.
00:40:20.370 --> 00:40:23.430
I'll explain to you how
it works in a second.
00:40:23.430 --> 00:40:25.650
So the formula is
f(x) turns out--
00:40:25.650 --> 00:40:28.560
There's a formula in terms
of the derivatives of f.
00:40:28.560 --> 00:40:31.260
Namely, you
differentiate n times,
00:40:31.260 --> 00:40:34.260
and you evaluate it at 0, and
you divide by n factorial,
00:40:34.260 --> 00:40:37.640
and multiply by x^n.
00:40:37.640 --> 00:40:40.510
So here's Taylor's formula.
00:40:40.510 --> 00:40:44.270
This tells you what
the Taylor series is.
00:40:44.270 --> 00:40:48.330
Now about half of our job
for the next few minutes
00:40:48.330 --> 00:40:51.630
is going to be to
give examples of this.
00:40:51.630 --> 00:40:56.810
But let me just explain
to you why this has to be.
00:40:56.810 --> 00:41:00.830
If you pick out this number
here, this is the a_n,
00:41:00.830 --> 00:41:03.290
the magic number a_n here.
00:41:03.290 --> 00:41:05.450
So let's just illustrate it.
00:41:05.450 --> 00:41:15.090
If f(x) happens to be a_0 + a_1
x + a_2 x^2 + a_3 x^3 plus dot,
00:41:15.090 --> 00:41:15.940
dot, dot.
00:41:15.940 --> 00:41:19.810
And now I differentiate
it, right?
00:41:19.810 --> 00:41:24.416
I get a 1 + 2 a_2 x + 3 a_3 x.
00:41:26.930 --> 00:41:33.120
If I differentiate it another
time, I get 2 a_2 plus 3--
00:41:33.120 --> 00:41:38.560
sorry, 3*2*a_3 x
plus dot, dot, dot.
00:41:38.560 --> 00:41:47.360
And now a third time, I
get 3*2 a_3 plus et cetera.
00:41:47.360 --> 00:41:54.000
So this next term is really in
disguise, 4*3*2 x a-- sorry,
00:41:54.000 --> 00:41:56.410
a_4 x.
00:41:56.410 --> 00:41:59.410
That's what really comes down if
I kept track of the fourth term
00:41:59.410 --> 00:42:00.980
there.
00:42:00.980 --> 00:42:04.660
So now here is my function.
00:42:04.660 --> 00:42:07.980
But now you see if
I plug in x = 0,
00:42:07.980 --> 00:42:13.750
I can pick off the third term.
00:42:13.750 --> 00:42:21.079
f triple prime of 0
is equal to 3*2 a_3.
00:42:21.079 --> 00:42:23.620
Right, because all the rest of
those terms, when I plug in 0,
00:42:23.620 --> 00:42:25.080
are just 0.
00:42:25.080 --> 00:42:26.730
Here's the formula.
00:42:26.730 --> 00:42:30.610
And so the pattern here is this.
00:42:30.610 --> 00:42:33.860
And what's really going on here
is this is really 3*2*1 a_3.
00:42:37.110 --> 00:42:48.480
And in general a_n is
equal to f, nth derivative,
00:42:48.480 --> 00:42:50.410
divided by n!.
00:42:50.410 --> 00:42:56.224
And of course, n!, I remind
you, is n times n-1 times
00:42:56.224 --> 00:42:57.570
n-2, all the way down to 1.
00:43:02.600 --> 00:43:05.980
Now there's one more
crazy convention
00:43:05.980 --> 00:43:08.220
which is always used.
00:43:08.220 --> 00:43:12.300
Which is that there's something
very strange here down at 0,
00:43:12.300 --> 00:43:16.990
which is that 0 factorial turns
out, has to be set equal to 1.
00:43:16.990 --> 00:43:19.270
All right, so that's
what you do in order
00:43:19.270 --> 00:43:20.540
to make this formula work out.
00:43:20.540 --> 00:43:22.623
And that's one of the
reasons for this convention.
00:43:29.470 --> 00:43:30.600
All right.
00:43:30.600 --> 00:43:36.680
So my next goal is to
give you some examples.
00:43:36.680 --> 00:43:45.280
And let's do a couple.
00:43:48.130 --> 00:43:51.880
So here's, well
you know, I'm going
00:43:51.880 --> 00:43:56.120
to have to let you see
a few of them next time.
00:43:56.120 --> 00:43:58.960
But let me just tell
you this one, which
00:43:58.960 --> 00:44:00.820
is by far the most impressive.
00:44:06.370 --> 00:44:11.840
So what happens with e^x -- if
the function is f(x) = e^x --
00:44:11.840 --> 00:44:18.190
is that its derivative
is also e^x.
00:44:18.190 --> 00:44:21.390
And its second
derivative is also e^x.
00:44:21.390 --> 00:44:23.010
And it just keeps
on going that way.
00:44:23.010 --> 00:44:25.180
They're all the same.
00:44:25.180 --> 00:44:30.010
So that means that these
numbers in Taylor's formula,
00:44:30.010 --> 00:44:33.020
in the numerator--
The nth derivative
00:44:33.020 --> 00:44:37.090
is very easy to evaluate.
00:44:37.090 --> 00:44:39.280
It's just e^x.
00:44:39.280 --> 00:44:43.890
And if I evaluate it
at x = 0, I just get 1.
00:44:43.890 --> 00:44:46.990
So all of those
numerators are 1.
00:44:46.990 --> 00:44:54.721
So the formula here, is the sum
n equals 0 to infinity, of 1/n!
00:44:54.721 --> 00:44:55.220
x^n.
00:45:00.960 --> 00:45:04.730
In particular, we now have
an honest formula for e
00:45:04.730 --> 00:45:06.800
to the first power.
00:45:06.800 --> 00:45:08.330
Which is just e.
00:45:08.330 --> 00:45:13.790
Which if I plug it in, x = 1, I
get 1, this is the n = 0 term.
00:45:13.790 --> 00:45:16.590
Plus 1, This is the n = 1 term.
00:45:16.590 --> 00:45:18.870
Plus 1/2!
00:45:18.870 --> 00:45:20.550
plus 1/3!
00:45:20.550 --> 00:45:21.150
plus 1/4!
00:45:27.040 --> 00:45:27.540
Right?
00:45:27.540 --> 00:45:31.360
So this is our first
honest formula for e.
00:45:31.360 --> 00:45:33.960
And also, this is
how you compute
00:45:33.960 --> 00:45:35.340
the exponential function.
00:45:41.750 --> 00:45:49.490
Finally if you take a
function like sin x,
00:45:49.490 --> 00:45:52.130
what you'll discover is that
we can complete the sort
00:45:52.130 --> 00:45:55.957
of strange business that we did
at the beginning of the course
00:45:55.957 --> 00:46:03.480
-- or cos x -- where we took
the linear and quadratic
00:46:03.480 --> 00:46:04.950
approximations.
00:46:04.950 --> 00:46:10.060
Now we're going to get complete
formulas for these functions.
00:46:10.060 --> 00:46:16.297
sin x turns out to be
equal to x - x^3 / 3!
00:46:16.297 --> 00:46:19.233
+ x^5 / 5!
00:46:19.233 --> 00:46:24.860
- x^7 / 7!, et cetera.
00:46:24.860 --> 00:46:30.680
And cos x = 1 - x^2 / 2!
00:46:30.680 --> 00:46:36.950
-- that's the same as this
2 here -- plus x^4 / 4!
00:46:36.950 --> 00:46:42.590
minus x^6 / 6!, plus et cetera.
00:46:42.590 --> 00:46:48.030
Now these may feel like they're
hard to memorize because I've
00:46:48.030 --> 00:46:49.425
just pulled them out of a hat.
00:46:51.990 --> 00:46:55.630
I do expect you to know them.
00:46:55.630 --> 00:46:58.990
They're actually extremely
similar formulas.
00:46:58.990 --> 00:47:02.230
The exponential here just has
this collection of factorials.
00:47:02.230 --> 00:47:06.860
The sine is all the odd
powers with alternating signs.
00:47:06.860 --> 00:47:10.260
And the cosine is all the even
powers with alternating signs.
00:47:10.260 --> 00:47:14.130
So all three of them form
part of the same family.
00:47:14.130 --> 00:47:16.140
So this will actually
make it easier
00:47:16.140 --> 00:47:18.150
for you to remember,
rather than harder.
00:47:21.490 --> 00:47:24.100
And so with that, I'll
leave the practice
00:47:24.100 --> 00:47:26.210
on differentiation
for next time.
00:47:26.210 --> 00:47:27.570
And good luck, everybody.
00:47:27.570 --> 00:47:29.940
I'll talk to you individually.