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PROFESSOR: Today we
are continuing

10
00:00:22,760 --> 00:00:24,730
with improper integrals.

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00:00:24,730 --> 00:00:27,371
I still have a little bit more
to tell you about them.

12
00:00:27,371 --> 00:00:30,080

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00:00:30,080 --> 00:00:35,250
What we were discussing at the
very end of last time was

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00:00:35,250 --> 00:00:36,500
improper integrals.

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00:00:36,500 --> 00:00:41,560

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00:00:41,560 --> 00:00:44,850
Now and these are going to be
improper integrals of the

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00:00:44,850 --> 00:00:48,220
second kind.

18
00:00:48,220 --> 00:00:51,270
By second kind I mean that they
have a singularity at a

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00:00:51,270 --> 00:00:52,520
finite place.

20
00:00:52,520 --> 00:00:54,800

21
00:00:54,800 --> 00:00:57,260
That would be something
like this.

22
00:00:57,260 --> 00:01:00,140
So here's the definition
if you like.

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00:01:00,140 --> 00:01:02,800
Same sort of thing as
we did when the

24
00:01:02,800 --> 00:01:04,120
singularity was at infinity.

25
00:01:04,120 --> 00:01:09,440
So if you have the integral
from 0 to 1 of f of x.

26
00:01:09,440 --> 00:01:12,600
This is going to be the same
thing as the limit.

27
00:01:12,600 --> 00:01:15,790
As a goes to 0 from above.

28
00:01:15,790 --> 00:01:21,150
The integral from a to
1 of f of x d x.

29
00:01:21,150 --> 00:01:25,920
And the idea here is the same
one that we had at infinity.

30
00:01:25,920 --> 00:01:27,320
Let me draw a picture of it.

31
00:01:27,320 --> 00:01:30,030
You have, imagine a function
which is coming down like this

32
00:01:30,030 --> 00:01:32,020
and here's the point 1.

33
00:01:32,020 --> 00:01:35,630
And we don't know whether the
area enclosed is going to be

34
00:01:35,630 --> 00:01:40,680
infinite or finite and so we
cut it off at some place a.

35
00:01:40,680 --> 00:01:44,220
And we let a go to
0 from above.

36
00:01:44,220 --> 00:01:46,800
So really it's 0 plus.

37
00:01:46,800 --> 00:01:49,820
So we're coming in from
the right here.

38
00:01:49,820 --> 00:01:52,810
And we're counting up the
area in this chunk.

39
00:01:52,810 --> 00:01:56,790
And we're seeing as it expands
whether it goes to infinity or

40
00:01:56,790 --> 00:01:59,290
whether it tends to
some finite limit.

41
00:01:59,290 --> 00:02:02,990
Right, so this is the example
and this is the definition.

42
00:02:02,990 --> 00:02:06,420
And just as we did for the
other kind of improper

43
00:02:06,420 --> 00:02:08,073
integral, we say that
this converges--

44
00:02:08,073 --> 00:02:11,160

45
00:02:11,160 --> 00:02:19,980
so that's the key word here
--if the limit is finite,

46
00:02:19,980 --> 00:02:26,390
exists maybe I should just
say and diverges if not.

47
00:02:26,390 --> 00:02:35,650

48
00:02:35,650 --> 00:02:38,945
Let's just take care of
the basic examples.

49
00:02:38,945 --> 00:02:42,110

50
00:02:42,110 --> 00:02:44,740
First of all I wrote this
one down last time.

51
00:02:44,740 --> 00:02:47,390
We're going to evaluate
this one.

52
00:02:47,390 --> 00:02:51,565
The integral from 0 to 1 of 1
over the square root of x.

53
00:02:51,565 --> 00:02:55,740

54
00:02:55,740 --> 00:02:57,690
And this just, you almost don't
notice the fact that it

55
00:02:57,690 --> 00:02:59,920
goes to infinity.

56
00:02:59,920 --> 00:03:02,570
This goes to infinity
as x goes to 0.

57
00:03:02,570 --> 00:03:04,320
But if you evaluate it --

58
00:03:04,320 --> 00:03:06,860
first of all we always write
this as a power.

59
00:03:06,860 --> 00:03:08,010
Right?

60
00:03:08,010 --> 00:03:09,870
To get the evaluation.

61
00:03:09,870 --> 00:03:13,060
And then I'm not even going
to replace the 0 by an a.

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00:03:13,060 --> 00:03:14,490
I'm just going to
leave it as 0.

63
00:03:14,490 --> 00:03:18,660
The antiderivative here is
x to the 1/2 times 2.

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00:03:18,660 --> 00:03:21,540

65
00:03:21,540 --> 00:03:23,750
And then I evaluate
that at 0 and 1.

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00:03:23,750 --> 00:03:25,080
And I get 2.

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00:03:25,080 --> 00:03:29,090
2 minus 0, which is 2.

68
00:03:29,090 --> 00:03:31,330
All right so this one
is convergent.

69
00:03:31,330 --> 00:03:34,180
And not only is it convergent
but we can evaluate it.

70
00:03:34,180 --> 00:03:38,310

71
00:03:38,310 --> 00:03:42,940
The second example being not
systematic but really giving

72
00:03:42,940 --> 00:03:46,990
you the principal examples that
we'll be thinking about.

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00:03:46,990 --> 00:03:53,920
Is this one here d x over x
And this one gives you the

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00:03:53,920 --> 00:03:56,070
antiderivative as
the logarithm.

75
00:03:56,070 --> 00:03:58,180
Evaluated at 0 and 1.

76
00:03:58,180 --> 00:04:00,910
And now again you have to have
this thought process in your

77
00:04:00,910 --> 00:04:02,770
mind that you're really
taking the limit.

78
00:04:02,770 --> 00:04:06,440
But this is going to be the log
of 1 minus the log of 0.

79
00:04:06,440 --> 00:04:07,830
Really the log of
0 from above.

80
00:04:07,830 --> 00:04:10,970
There is no such thing as
the log of 0 from below.

81
00:04:10,970 --> 00:04:12,820
And this is minus infinity.

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00:04:12,820 --> 00:04:16,000
So it's 0 minus minus infinity,

83
00:04:16,000 --> 00:04:19,020
which is plus infinity.

84
00:04:19,020 --> 00:04:20,270
And so this one diverges.

85
00:04:20,270 --> 00:04:29,710

86
00:04:29,710 --> 00:04:32,750
All right so what's
the general?

87
00:04:32,750 --> 00:04:39,070
So more or less in general,
let's just, for powers anyway,

88
00:04:39,070 --> 00:04:43,097
if you work out this thing
for d x over x to

89
00:04:43,097 --> 00:04:45,220
the p from 0 to 1.

90
00:04:45,220 --> 00:04:50,150
What you're going to find is
that it's 1 over 1 minus p.

91
00:04:50,150 --> 00:04:53,020
When p is less than one.

92
00:04:53,020 --> 00:05:00,240
And it diverges for p bigger
than or equal to 1.

93
00:05:00,240 --> 00:05:02,750

94
00:05:02,750 --> 00:05:05,310
Now that's the final result.

95
00:05:05,310 --> 00:05:06,630
If you carry out this

96
00:05:06,630 --> 00:05:10,920
integration it's not difficult.

97
00:05:10,920 --> 00:05:15,170
All right so now I just
want to try to help

98
00:05:15,170 --> 00:05:17,270
you to remember this.

99
00:05:17,270 --> 00:05:20,500
And to think about how you
should think about it.

100
00:05:20,500 --> 00:05:23,890
So I'm going to say it
in a few more ways.

101
00:05:23,890 --> 00:05:27,890
All right just repeat what I've
said already but try to

102
00:05:27,890 --> 00:05:32,020
get it to percolate
and absorb itself.

103
00:05:32,020 --> 00:05:36,190
And in order to do that I have
to make the contrast between

104
00:05:36,190 --> 00:05:38,070
the kind of improper
integral that I was

105
00:05:38,070 --> 00:05:39,410
dealing with before.

106
00:05:39,410 --> 00:05:43,230
Which was not as x goes to
0 here but as x goes to

107
00:05:43,230 --> 00:05:45,920
infinity, the other side.

108
00:05:45,920 --> 00:05:52,470
Let's make this contrast.

109
00:05:52,470 --> 00:05:56,410
First of all, if I look at the
angle that we have been paying

110
00:05:56,410 --> 00:05:57,610
attention to right now.

111
00:05:57,610 --> 00:06:00,580
We've just considered
things like this.

112
00:06:00,580 --> 00:06:02,520
1 over x to the 1 half.

113
00:06:02,520 --> 00:06:06,600
Which is a lot smaller
than one over x.

114
00:06:06,600 --> 00:06:10,740
Which is a lot smaller than
say 1 over x squared.

115
00:06:10,740 --> 00:06:12,050
Which would be another
example.

116
00:06:12,050 --> 00:06:14,560
This is as x goes to 0.

117
00:06:14,560 --> 00:06:19,990

118
00:06:19,990 --> 00:06:22,280
So this one's the
smallest one.

119
00:06:22,280 --> 00:06:23,310
This one's the next
smallest one.

120
00:06:23,310 --> 00:06:26,820
And this one is very large.

121
00:06:26,820 --> 00:06:30,985
On the other hand it goes the
other way at infinity.

122
00:06:30,985 --> 00:06:36,670

123
00:06:36,670 --> 00:06:39,936
As x tends to infinity.

124
00:06:39,936 --> 00:06:43,140
All right so try to
keep that in mind.

125
00:06:43,140 --> 00:06:47,910
And now I'm going to put
a little box around

126
00:06:47,910 --> 00:06:50,100
the bad guys here.

127
00:06:50,100 --> 00:06:54,710
This one is divergent.

128
00:06:54,710 --> 00:06:57,870
And this one is divergent.

129
00:06:57,870 --> 00:06:59,950
And this one is divergent.

130
00:06:59,950 --> 00:07:01,020
And this one is divergent.

131
00:07:01,020 --> 00:07:03,510
When the cross over
point is 1 over x.

132
00:07:03,510 --> 00:07:06,410
When we get smaller than that,
we get to things which are

133
00:07:06,410 --> 00:07:07,390
convergent.

134
00:07:07,390 --> 00:07:10,820
When we get smaller than it
on this other scale, it's

135
00:07:10,820 --> 00:07:12,680
convergent.

136
00:07:12,680 --> 00:07:13,995
All right so these guys
are divergent.

137
00:07:13,995 --> 00:07:20,300

138
00:07:20,300 --> 00:07:23,430
So they associated with
divergent integrals.

139
00:07:23,430 --> 00:07:26,390
The functions themselves are
just tending towards-- well

140
00:07:26,390 --> 00:07:29,310
these tend to infinity, and
these tend toward 0.

141
00:07:29,310 --> 00:07:33,320
So I'm not talking about
the functions

142
00:07:33,320 --> 00:07:35,190
themselves but the integrals.

143
00:07:35,190 --> 00:07:40,040
Now I want to draw this again
here, not small enough.

144
00:07:40,040 --> 00:07:43,810

145
00:07:43,810 --> 00:07:45,060
I want to draw this again.

146
00:07:45,060 --> 00:07:48,980

147
00:07:48,980 --> 00:07:50,800
And, so I'm just going to draw
a picture of what it

148
00:07:50,800 --> 00:07:51,760
is that I have here.

149
00:07:51,760 --> 00:07:54,850
But I'm going to combine
these two pictures.

150
00:07:54,850 --> 00:08:01,630
So here's the picture for
example of y equals 1 over x.

151
00:08:01,630 --> 00:08:04,714

152
00:08:04,714 --> 00:08:06,700
All right.

153
00:08:06,700 --> 00:08:08,770
That's y equals 1 over x.

154
00:08:08,770 --> 00:08:10,470
And that picture is
very balanced.

155
00:08:10,470 --> 00:08:12,600
It's symmetric on
the two ends.

156
00:08:12,600 --> 00:08:17,660
If I cut it in half then what
I get here is 2 halves.

157
00:08:17,660 --> 00:08:23,700
And this one has an
infinite area.

158
00:08:23,700 --> 00:08:27,060
That corresponds to the integral
from 1 to infinity, d

159
00:08:27,060 --> 00:08:30,530
x over x being infinite.

160
00:08:30,530 --> 00:08:32,940
And the other piece--

161
00:08:32,940 --> 00:08:35,310
which this one we calculated
last time.

162
00:08:35,310 --> 00:08:37,300
This is the one that we just
calculated over here at

163
00:08:37,300 --> 00:08:42,719
example 2 has the
same property.

164
00:08:42,719 --> 00:08:45,720
It's infinite.

165
00:08:45,720 --> 00:08:48,102
And that's the fact that the
integral from 0 to 1 of dx

166
00:08:48,102 --> 00:08:52,100
over x is infinite.

167
00:08:52,100 --> 00:08:56,570
Right, so both, we lose
on both ends.

168
00:08:56,570 --> 00:09:02,390
On the other hand if I
take something like--

169
00:09:02,390 --> 00:09:05,340
I'm drawing it the same way but
it's really not the same

170
00:09:05,340 --> 00:09:09,540
--y equals 1 over the square
root of x. y equals 1 over x

171
00:09:09,540 --> 00:09:11,010
to the 1 half.

172
00:09:11,010 --> 00:09:17,620
And if I cut that in half here
then the x to the 1 half is

173
00:09:17,620 --> 00:09:19,890
actually bigger than this guy.

174
00:09:19,890 --> 00:09:21,740
So this piece is infinite.

175
00:09:21,740 --> 00:09:26,830

176
00:09:26,830 --> 00:09:29,930
And this part over here actually
is going to give us

177
00:09:29,930 --> 00:09:31,310
an honest number.

178
00:09:31,310 --> 00:09:34,810
In fact this one is finite.

179
00:09:34,810 --> 00:09:36,550
And we just checked what
the number is.

180
00:09:36,550 --> 00:09:38,340
It actually happens
to have area 2.

181
00:09:38,340 --> 00:09:46,250

182
00:09:46,250 --> 00:09:49,570
And what's happening here is if
you would superimpose this

183
00:09:49,570 --> 00:09:52,130
graph on the other graph
what you would see

184
00:09:52,130 --> 00:09:54,580
is that they cross.

185
00:09:54,580 --> 00:09:58,970
And this one sits on top.

186
00:09:58,970 --> 00:10:04,660
So if I drew this one in let's
have another color here

187
00:10:04,660 --> 00:10:05,840
--orange let's say.

188
00:10:05,840 --> 00:10:08,790
If this were orange if I
set it on top here it

189
00:10:08,790 --> 00:10:11,130
would go this way.

190
00:10:11,130 --> 00:10:14,790
OK and underneath the orange
is still infinite.

191
00:10:14,790 --> 00:10:16,070
So both of these are infinite.

192
00:10:16,070 --> 00:10:18,250
On here on the other hand
underneath the orange is

193
00:10:18,250 --> 00:10:21,700
infinite but underneath where
the green is finite.

194
00:10:21,700 --> 00:10:23,770
That's a smaller quantity.

195
00:10:23,770 --> 00:10:25,290
Infinity is a lot
bigger than 2.

196
00:10:25,290 --> 00:10:27,826
2 is a lot less than infinity.

197
00:10:27,826 --> 00:10:30,835
All right so that's reflected
in these comparisons here.

198
00:10:30,835 --> 00:10:33,610
Now if you like if I want
to do these in green.

199
00:10:33,610 --> 00:10:39,900
This guy is good and
this guy is good.

200
00:10:39,900 --> 00:10:43,050
Well let me just repeat that
idea over here in this sort of

201
00:10:43,050 --> 00:10:47,930
reversed picture with y equals
1 over x squared.

202
00:10:47,930 --> 00:10:50,910
If I chop that in half
then the good

203
00:10:50,910 --> 00:10:53,640
part is this end here.

204
00:10:53,640 --> 00:10:54,890
This is finite.

205
00:10:54,890 --> 00:10:56,990

206
00:10:56,990 --> 00:10:59,570
And the bad part is
this part of here

207
00:10:59,570 --> 00:11:01,480
which is way more singular.

208
00:11:01,480 --> 00:11:02,730
And it's infinite.

209
00:11:02,730 --> 00:11:07,070

210
00:11:07,070 --> 00:11:10,750
All right so again what I've
just tried to do is to give

211
00:11:10,750 --> 00:11:18,580
you some geometric sense and
also some visceral sense.

212
00:11:18,580 --> 00:11:23,040
This guy it's tail as it goes
out to infinity is much lower.

213
00:11:23,040 --> 00:11:25,470
It's much smaller
than 1 over x.

214
00:11:25,470 --> 00:11:28,080
And these guys trapped an
infinite amount of area.

215
00:11:28,080 --> 00:11:30,110
This one traps only a finite
amount of area.

216
00:11:30,110 --> 00:11:36,676

217
00:11:36,676 --> 00:11:40,650
All right so now I'm just going
to give one last example

218
00:11:40,650 --> 00:11:43,090
which combines these two
types of pictures.

219
00:11:43,090 --> 00:11:45,480
It's really practically
the same as what I've

220
00:11:45,480 --> 00:11:47,370
said before but I--

221
00:11:47,370 --> 00:11:50,340

222
00:11:50,340 --> 00:11:53,320
oh have to erase this one too.

223
00:11:53,320 --> 00:12:01,340

224
00:12:01,340 --> 00:12:07,330
So here's another example if
you're in so let's take the

225
00:12:07,330 --> 00:12:08,160
following example.

226
00:12:08,160 --> 00:12:09,840
This is somewhat related
to the first one

227
00:12:09,840 --> 00:12:11,600
that I gave last time.

228
00:12:11,600 --> 00:12:15,700
If you take a function
y equals 1

229
00:12:15,700 --> 00:12:18,770
over x minus 3 squared.

230
00:12:18,770 --> 00:12:21,160
And you think about
it's integral.

231
00:12:21,160 --> 00:12:24,750
So let's think about the
integral from 0 to infinity, d

232
00:12:24,750 --> 00:12:27,090
x over x minus 3 squared.

233
00:12:27,090 --> 00:12:30,420
And suppose you were faced
with this integral.

234
00:12:30,420 --> 00:12:33,210
In order to understand what
it's doing you have to pay

235
00:12:33,210 --> 00:12:36,310
attention to two places
where it can go wrong.

236
00:12:36,310 --> 00:12:39,530
We're going to split
into two pieces.

237
00:12:39,530 --> 00:12:44,730
I'm going say break it up into
this one here up to 5, for the

238
00:12:44,730 --> 00:12:45,980
sake of argument.

239
00:12:45,980 --> 00:12:48,090

240
00:12:48,090 --> 00:12:49,715
And say from 5 to infinity.

241
00:12:49,715 --> 00:12:53,672

242
00:12:53,672 --> 00:12:54,960
All right.

243
00:12:54,960 --> 00:12:56,260
So these are the two chunks.

244
00:12:56,260 --> 00:12:58,810
Now why did I break it up
into those two pieces?

245
00:12:58,810 --> 00:13:01,890
Because what's happening with
this function is that it's

246
00:13:01,890 --> 00:13:05,250
going up like this at 3.

247
00:13:05,250 --> 00:13:08,470
And so if I look at the
two halves here.

248
00:13:08,470 --> 00:13:10,160
I'm going to draw them again
and I'm going to illustrate

249
00:13:10,160 --> 00:13:12,560
them with the colors
we've chosen--

250
00:13:12,560 --> 00:13:16,350
which are I guess red and green
--what you'll discover

251
00:13:16,350 --> 00:13:28,350
is that this one here, which
corresponds to this piece here

252
00:13:28,350 --> 00:13:30,540
is infinite.

253
00:13:30,540 --> 00:13:33,250
And it's infinite because
there's a square in the

254
00:13:33,250 --> 00:13:34,270
denominator.

255
00:13:34,270 --> 00:13:39,560
And as x goes to 3 this is very
much like if we shifted

256
00:13:39,560 --> 00:13:40,670
the 3 to 0.

257
00:13:40,670 --> 00:13:42,560
Very much like this 1
over x squared here.

258
00:13:42,560 --> 00:13:44,250
But not in this context.

259
00:13:44,250 --> 00:13:46,330
In the other context where
it's going to infinity.

260
00:13:46,330 --> 00:13:49,460

261
00:13:49,460 --> 00:13:54,320
This is the same as at the
picture directly above with

262
00:13:54,320 --> 00:13:57,398
the infinite part in red.

263
00:13:57,398 --> 00:13:59,650
All right.

264
00:13:59,650 --> 00:14:04,915
And this part here, this
part is finite.

265
00:14:04,915 --> 00:14:06,240
All right.

266
00:14:06,240 --> 00:14:09,050
So since we have an infinite
part plus a finite part the

267
00:14:09,050 --> 00:14:15,070
conclusion is that this thing,
well this guy converges.

268
00:14:15,070 --> 00:14:18,850
And this one diverges.

269
00:14:18,850 --> 00:14:21,470

270
00:14:21,470 --> 00:14:23,930
But the total unfortunately
diverges.

271
00:14:23,930 --> 00:14:25,760
Right because it's got
1 infinity in it.

272
00:14:25,760 --> 00:14:28,370
So this thing diverges.

273
00:14:28,370 --> 00:14:31,990

274
00:14:31,990 --> 00:14:33,480
And that's what happened
last time when

275
00:14:33,480 --> 00:14:34,870
we got a crazy number.

276
00:14:34,870 --> 00:14:36,150
If you integrated this
you would get

277
00:14:36,150 --> 00:14:37,480
some negative number.

278
00:14:37,480 --> 00:14:39,850
If you wrote down the
formulas carelessly.

279
00:14:39,850 --> 00:14:41,320
And the reason is that the

280
00:14:41,320 --> 00:14:44,710
calculation actually is nonsense.

281
00:14:44,710 --> 00:14:48,590
So you've gotta be aware if you
encounter a singularity in

282
00:14:48,590 --> 00:14:51,840
the middle not to ignore it.

283
00:14:51,840 --> 00:14:52,100
Yeah.

284
00:14:52,100 --> 00:14:52,620
Question.

285
00:14:52,620 --> 00:14:53,870
AUDIENCE: [INAUDIBLE PHRASE]

286
00:14:53,870 --> 00:14:56,430

287
00:14:56,430 --> 00:14:59,710
PROFESSOR: Why do we say that
the whole thing diverges?

288
00:14:59,710 --> 00:15:02,490
The reason why we say that is
the area under the whole curve

289
00:15:02,490 --> 00:15:03,690
is infinite.

290
00:15:03,690 --> 00:15:06,130
It's the sum of this piece
plus this piece.

291
00:15:06,130 --> 00:15:08,886
And so the total is infinite.

292
00:15:08,886 --> 00:15:10,136
AUDIENCE: [INAUDIBLE PHRASE]

293
00:15:10,136 --> 00:15:17,370

294
00:15:17,370 --> 00:15:17,990
PROFESSOR: We're stuck.

295
00:15:17,990 --> 00:15:19,270
This is an ill defined
integral.

296
00:15:19,270 --> 00:15:21,720
It's one where you're
red flashing warning

297
00:15:21,720 --> 00:15:22,560
sign should be on.

298
00:15:22,560 --> 00:15:24,140
Because you're not going to
get the right answer by

299
00:15:24,140 --> 00:15:24,510
computing it.

300
00:15:24,510 --> 00:15:26,860
You'll never get an answer.

301
00:15:26,860 --> 00:15:29,230
Similarly you'll never get
an answer with this.

302
00:15:29,230 --> 00:15:32,730
And you will get an
answer with that.

303
00:15:32,730 --> 00:15:33,980
OK?

304
00:15:33,980 --> 00:15:37,220

305
00:15:37,220 --> 00:15:39,640
Yeah another question.

306
00:15:39,640 --> 00:15:40,890
AUDIENCE: [INAUDIBLE PHRASE]

307
00:15:40,890 --> 00:15:45,760

308
00:15:45,760 --> 00:15:49,480
PROFESSOR: So the question is if
you have a little glance at

309
00:15:49,480 --> 00:15:52,930
an integral, how are you
going to decide where

310
00:15:52,930 --> 00:15:54,750
you should be heading?

311
00:15:54,750 --> 00:15:58,280
So I'm going to answer
that orally.

312
00:15:58,280 --> 00:16:04,010
Although you know, but I'll
say one little hint here.

313
00:16:04,010 --> 00:16:08,290
So you always have to check x
going to infinity and x going

314
00:16:08,290 --> 00:16:10,740
to minus infinity, if
they're in there.

315
00:16:10,740 --> 00:16:15,460
And you also have to check any
singularity, like x going to 3

316
00:16:15,460 --> 00:16:16,780
for sure in this case.

317
00:16:16,780 --> 00:16:18,330
You have to pay attention
all the places where

318
00:16:18,330 --> 00:16:19,350
the thing is infinite.

319
00:16:19,350 --> 00:16:22,330
And then you want to focus in
on each one separately.

320
00:16:22,330 --> 00:16:26,920
And decide what's going on it
at that particular place.

321
00:16:26,920 --> 00:16:29,590
When it's a negative power.

322
00:16:29,590 --> 00:16:36,980
So remember d x over x as
x goes to 0 is bad.

323
00:16:36,980 --> 00:16:39,070
And d x over x squared is bad.

324
00:16:39,070 --> 00:16:40,490
D x over x cubed is bad.

325
00:16:40,490 --> 00:16:42,600
All of them are even worse.

326
00:16:42,600 --> 00:16:49,650
So anything of this form is
bad and equals 1, 2, 3.

327
00:16:49,650 --> 00:16:51,980
These are the red box kinds.

328
00:16:51,980 --> 00:16:55,030
All right.

329
00:16:55,030 --> 00:16:58,220
That means that any of the
integrals that we did in

330
00:16:58,220 --> 00:17:01,890
partial fractions which had a
root, which had a factor of

331
00:17:01,890 --> 00:17:03,160
something, the denominator.

332
00:17:03,160 --> 00:17:05,480
Those are all divergent
integrals if you cross the

333
00:17:05,480 --> 00:17:06,670
singularly.

334
00:17:06,670 --> 00:17:09,357
Not a single one of them makes
sense across the singularity.

335
00:17:09,357 --> 00:17:10,607
Right?

336
00:17:10,607 --> 00:17:12,750

337
00:17:12,750 --> 00:17:14,850
If you have square roots and
things like that then you can

338
00:17:14,850 --> 00:17:16,100
repair things like that.

339
00:17:16,100 --> 00:17:18,030
And there's some interesting
examples of that.

340
00:17:18,030 --> 00:17:21,080
Such as with the arc sign
function and so forth.

341
00:17:21,080 --> 00:17:25,940
Where you have an improper
integral which is really OK.

342
00:17:25,940 --> 00:17:26,730
All right.

343
00:17:26,730 --> 00:17:29,880
So that's the best I can do.

344
00:17:29,880 --> 00:17:32,260
It's obviously something you
get experience with.

345
00:17:32,260 --> 00:17:34,110
All right.

346
00:17:34,110 --> 00:17:38,550
Now I'm going to move on
and this is more or

347
00:17:38,550 --> 00:17:42,150
less our last topic.

348
00:17:42,150 --> 00:17:43,900
Yay, but not quite.

349
00:17:43,900 --> 00:17:46,730
Well, so I should say it's
our penultimate topic.

350
00:17:46,730 --> 00:17:49,430
Right because we have
one more lecture.

351
00:17:49,430 --> 00:17:52,010
All right.

352
00:17:52,010 --> 00:17:54,750
So that our next topic
is series.

353
00:17:54,750 --> 00:17:58,810
Now we'll do it in a sort
of a concrete way today.

354
00:17:58,810 --> 00:18:01,140
And then we'll do what
are known as

355
00:18:01,140 --> 00:18:02,775
power series tomorrow.

356
00:18:02,775 --> 00:18:05,400

357
00:18:05,400 --> 00:18:06,760
So let me tell you
about series.

358
00:18:06,760 --> 00:18:20,490

359
00:18:20,490 --> 00:18:22,790
Remember we're talking
about infinity and

360
00:18:22,790 --> 00:18:24,040
dealing with infinity.

361
00:18:24,040 --> 00:18:26,820

362
00:18:26,820 --> 00:18:28,730
So we're not just talking
about any old series.

363
00:18:28,730 --> 00:18:30,130
We're talking about
infinite series.

364
00:18:30,130 --> 00:18:32,650

365
00:18:32,650 --> 00:18:37,690
There is one infinite series
which is probably, which is

366
00:18:37,690 --> 00:18:41,980
without question the most
important and useful series.

367
00:18:41,980 --> 00:18:44,870
And that's the geometric
series but I'm going to

368
00:18:44,870 --> 00:18:48,405
introduce it concretely first
in a particular case.

369
00:18:48,405 --> 00:18:51,990

370
00:18:51,990 --> 00:18:54,260
If I draw a picture
of this sum.

371
00:18:54,260 --> 00:18:56,480
Which in principle
goes on forever.

372
00:18:56,480 --> 00:18:59,920
You can see that it goes
someplace fairly easily by

373
00:18:59,920 --> 00:19:02,730
marking out what's happening
on the number line.

374
00:19:02,730 --> 00:19:06,630
The first step takes
us to 1 from 0.

375
00:19:06,630 --> 00:19:11,790
And then if I add this half,
I get to 3 halves.

376
00:19:11,790 --> 00:19:14,130
Right, so the first
step was 1 and the

377
00:19:14,130 --> 00:19:16,240
second step was a half.

378
00:19:16,240 --> 00:19:21,060
Now if I add this quarter i,
which is the next piece then I

379
00:19:21,060 --> 00:19:22,400
get some place here.

380
00:19:22,400 --> 00:19:29,000
But what I want to observe is
that I got, I can look at it

381
00:19:29,000 --> 00:19:31,130
from the other point of view.

382
00:19:31,130 --> 00:19:32,500
I got, when I move this
quarter I got

383
00:19:32,500 --> 00:19:36,240
half way to 2 here.

384
00:19:36,240 --> 00:19:40,300
I'm putting 2 in green because
I want you to think of it as

385
00:19:40,300 --> 00:19:42,950
being the good kind.

386
00:19:42,950 --> 00:19:43,970
Right.

387
00:19:43,970 --> 00:19:45,420
The kind that has a number.

388
00:19:45,420 --> 00:19:47,280
And not one of the red kinds.

389
00:19:47,280 --> 00:19:49,920
We're getting there and
we're almost there.

390
00:19:49,920 --> 00:19:52,620
So the next stage we
get half way again.

391
00:19:52,620 --> 00:19:54,605
That's the eighth
and so forth.

392
00:19:54,605 --> 00:19:56,490
And eventually we get to 2.

393
00:19:56,490 --> 00:20:00,476
So this sum we write
equals two.

394
00:20:00,476 --> 00:20:02,940
All right that's kind of
a paradox because we

395
00:20:02,940 --> 00:20:04,130
never get to 2.

396
00:20:04,130 --> 00:20:08,080
This is the paradox that
Zeno fussed with.

397
00:20:08,080 --> 00:20:12,960
And his conclusion, you know,
with the rabbit and the hare.

398
00:20:12,960 --> 00:20:14,140
Oh no the rabbit and
the tortoise.

399
00:20:14,140 --> 00:20:16,120
Sorry hare chasing--

400
00:20:16,120 --> 00:20:18,890
anyway the rabbit chasing
the tortoise.

401
00:20:18,890 --> 00:20:21,720
His conclusion-- you know, I
don't know if you're aware of

402
00:20:21,720 --> 00:20:23,770
this --but he understood
this paradox.

403
00:20:23,770 --> 00:20:25,740
And he said you know it doesn't
look like it ever gets

404
00:20:25,740 --> 00:20:29,670
there because they're infinitely
many times between

405
00:20:29,670 --> 00:20:32,330
the time-- you know that the
tortoise is always behind,

406
00:20:32,330 --> 00:20:34,580
always behind, always behind,
always behind.

407
00:20:34,580 --> 00:20:36,570
So therefore it's impossible
that the

408
00:20:36,570 --> 00:20:38,710
tortoise catches up right.

409
00:20:38,710 --> 00:20:41,600
So do you know what his
conclusion was?

410
00:20:41,600 --> 00:20:46,550
Time does not exist. That was
actually literally his

411
00:20:46,550 --> 00:20:48,260
conclusion.

412
00:20:48,260 --> 00:20:49,950
Because he didn't understand
the possibility of a

413
00:20:49,950 --> 00:20:51,030
continuum of time.

414
00:20:51,030 --> 00:20:53,090
Because there were infinitely
many things that happened

415
00:20:53,090 --> 00:20:56,610
before the tortoise caught up.

416
00:20:56,610 --> 00:20:57,790
So that was the reasoning.

417
00:20:57,790 --> 00:21:00,430
I mean it's a long time ago but
you know people didn't, he

418
00:21:00,430 --> 00:21:02,430
didn't believe in continuum.

419
00:21:02,430 --> 00:21:03,480
All right.

420
00:21:03,480 --> 00:21:06,940
So anyway that's
a small point.

421
00:21:06,940 --> 00:21:19,890
Now the general case here of a
geometric series is where I

422
00:21:19,890 --> 00:21:22,820
put in a number a instead
of a half here.

423
00:21:22,820 --> 00:21:23,870
So what we had before.

424
00:21:23,870 --> 00:21:26,890
So that's 1 plus a
plus a squared.

425
00:21:26,890 --> 00:21:28,390
Isn't quite the most
general but anyway

426
00:21:28,390 --> 00:21:31,900
I'll write this down.

427
00:21:31,900 --> 00:21:34,720
And you're certainly going to
want to remember that the

428
00:21:34,720 --> 00:21:39,510
formula for this in the limit
is 1 over 1 minus a.

429
00:21:39,510 --> 00:21:43,470
And I remind you that this only
works when the absolute

430
00:21:43,470 --> 00:21:45,810
value is strictly less than 1.

431
00:21:45,810 --> 00:21:47,970
In other words when minus 1
is strictly less than a

432
00:21:47,970 --> 00:21:51,130
is less than 1.

433
00:21:51,130 --> 00:21:53,430
And that's really the issue that
we're going to want to

434
00:21:53,430 --> 00:21:54,310
worry about now.

435
00:21:54,310 --> 00:21:58,270
What we're worrying about is
this notion of convergence.

436
00:21:58,270 --> 00:22:06,130
And what goes wrong when there
isn't convergence, when

437
00:22:06,130 --> 00:22:07,360
there's a divergence.

438
00:22:07,360 --> 00:22:13,220
So let me illustrate the
divergences before going on.

439
00:22:13,220 --> 00:22:16,090
And this is what we have to
avoid if we're going to

440
00:22:16,090 --> 00:22:18,990
understand series.

441
00:22:18,990 --> 00:22:21,890
So here's an example when
a is equal to 1.

442
00:22:21,890 --> 00:22:26,620
You get 1 plus 1 plus
1 plus et cetera.

443
00:22:26,620 --> 00:22:29,990
And that's equal to
1 over 1 minus 1.

444
00:22:29,990 --> 00:22:32,370
Which is 1 over 0.

445
00:22:32,370 --> 00:22:33,760
So this is not bad.

446
00:22:33,760 --> 00:22:34,965
It's almost right.

447
00:22:34,965 --> 00:22:35,360
Right?

448
00:22:35,360 --> 00:22:37,750
It's sort of infinity
equals infinity.

449
00:22:37,750 --> 00:22:40,560
At the edge here we managed to
get something which is sort of

450
00:22:40,560 --> 00:22:42,340
almost right.

451
00:22:42,340 --> 00:22:46,100
But you know, it's, we don't
consider this to be logically

452
00:22:46,100 --> 00:22:47,540
to make complete sense.

453
00:22:47,540 --> 00:22:51,200
So it's a little dangerous.

454
00:22:51,200 --> 00:22:52,960
And so we just say
that it diverges.

455
00:22:52,960 --> 00:22:54,100
And we get rid of this.

456
00:22:54,100 --> 00:22:55,990
So we're still putting
it in red.

457
00:22:55,990 --> 00:22:58,540
All right.

458
00:22:58,540 --> 00:23:00,110
The bad guy here.

459
00:23:00,110 --> 00:23:01,360
So this one diverges.

460
00:23:01,360 --> 00:23:04,670

461
00:23:04,670 --> 00:23:10,710
Similarly if I take a equals
minus 1, I get 1 minus 1 plus

462
00:23:10,710 --> 00:23:14,610
1 minus one plus 1 because the
odd and the even powers in

463
00:23:14,610 --> 00:23:17,270
that formula alternate sign.

464
00:23:17,270 --> 00:23:19,750
And this bounces
back and forth.

465
00:23:19,750 --> 00:23:21,640
It never settles down.

466
00:23:21,640 --> 00:23:23,560
It starts at 1.

467
00:23:23,560 --> 00:23:25,720
And then it gets down to 0 and
then it goes back up to 1,

468
00:23:25,720 --> 00:23:28,400
down to 0, back up to 1.

469
00:23:28,400 --> 00:23:29,570
It doesn't settle down.

470
00:23:29,570 --> 00:23:30,600
It bounces back and forth.

471
00:23:30,600 --> 00:23:31,580
It oscillates.

472
00:23:31,580 --> 00:23:34,740
On the other hand if you compare
the right hand side.

473
00:23:34,740 --> 00:23:35,980
What's the right hand side?

474
00:23:35,980 --> 00:23:39,730
It's 1 over 1 minus minus 1.

475
00:23:39,730 --> 00:23:41,515
which is a half.

476
00:23:41,515 --> 00:23:42,460
All right.

477
00:23:42,460 --> 00:23:45,120
So if you just paid attention
to the formula.

478
00:23:45,120 --> 00:23:47,240
Which is what we were doing
when we integrated without

479
00:23:47,240 --> 00:23:49,240
thinking too hard about this.

480
00:23:49,240 --> 00:23:51,320
You get a number here but
in fact that's wrong.

481
00:23:51,320 --> 00:23:52,830
Actually it's kind of an
interesting number.

482
00:23:52,830 --> 00:23:56,380
It's halfway between the
two between 0 1.

483
00:23:56,380 --> 00:23:59,570
So again there's some sort of
vague sense in which this is

484
00:23:59,570 --> 00:24:01,966
trying to be this answer.

485
00:24:01,966 --> 00:24:04,580
All right.

486
00:24:04,580 --> 00:24:06,730
It's not so bad but we're
still going to put

487
00:24:06,730 --> 00:24:08,744
this in a red box.

488
00:24:08,744 --> 00:24:10,030
All right.

489
00:24:10,030 --> 00:24:12,710
because this is what we
called divergence.

490
00:24:12,710 --> 00:24:16,130
So both of these cases
are divergent.

491
00:24:16,130 --> 00:24:20,490
It only really works when alpha,
when a is less than 1.

492
00:24:20,490 --> 00:24:25,170
I'm going to add one more
case just to see that

493
00:24:25,170 --> 00:24:31,550
mathematicians are slightly
curious about what goes on in

494
00:24:31,550 --> 00:24:32,020
other cases.

495
00:24:32,020 --> 00:24:35,130
So this is 1 plus 2 plus
2 squared plus 2

496
00:24:35,130 --> 00:24:37,570
to cube plus etc..

497
00:24:37,570 --> 00:24:39,960
And that should be equal to--

498
00:24:39,960 --> 00:24:44,750
according to this formula
--1 over 1 minus 2.

499
00:24:44,750 --> 00:24:48,242
Which is negative 1.

500
00:24:48,242 --> 00:24:49,860
All right.

501
00:24:49,860 --> 00:24:53,460
Now this one it clearly
wrong, right?

502
00:24:53,460 --> 00:24:55,530
This one is totally wrong.

503
00:24:55,530 --> 00:24:58,170

504
00:24:58,170 --> 00:24:59,465
It certainly diverges.

505
00:24:59,465 --> 00:25:02,370
The left hand side is
obviously infinite.

506
00:25:02,370 --> 00:25:04,070
The right hand side
is way off.

507
00:25:04,070 --> 00:25:05,960
It's negative 1.

508
00:25:05,960 --> 00:25:11,330
On the other hand it turns out
actually that mathematicians

509
00:25:11,330 --> 00:25:13,360
have ways of making sense
out of these.

510
00:25:13,360 --> 00:25:16,110
In number theory there's a
strange system where this is

511
00:25:16,110 --> 00:25:18,160
actually true.

512
00:25:18,160 --> 00:25:22,070
And what happens in that system
is that what you have

513
00:25:22,070 --> 00:25:27,050
to throw out is the idea
that 0 is less than 1.

514
00:25:27,050 --> 00:25:29,980
There is no such thing
as negative numbers.

515
00:25:29,980 --> 00:25:32,090
So this number exists.

516
00:25:32,090 --> 00:25:35,700
And it's the additive
inverse of 1.

517
00:25:35,700 --> 00:25:39,780
It has this arithmetic property
but the statement

518
00:25:39,780 --> 00:25:43,330
that this is, that 1 is bigger
than 0 does not make sense.

519
00:25:43,330 --> 00:25:45,830
So you have your choice either
this diverges or you have to

520
00:25:45,830 --> 00:25:48,630
throw out something like this.

521
00:25:48,630 --> 00:25:51,510
So that's a very curious thing
in higher mathematics.

522
00:25:51,510 --> 00:25:54,070
Which if you get to
number theory

523
00:25:54,070 --> 00:25:56,410
there's fun stuff there.

524
00:25:56,410 --> 00:25:58,920
All right.

525
00:25:58,920 --> 00:26:02,740
OK but for our purposes these
things are all out.

526
00:26:02,740 --> 00:26:03,300
All right.

527
00:26:03,300 --> 00:26:04,010
They're gone.

528
00:26:04,010 --> 00:26:05,160
We're not considering them.

529
00:26:05,160 --> 00:26:09,550
Only a between negative
1 and 1.

530
00:26:09,550 --> 00:26:10,800
All right.

531
00:26:10,800 --> 00:26:13,910

532
00:26:13,910 --> 00:26:18,190
Now I want to do something
systematic.

533
00:26:18,190 --> 00:26:21,870
And it's more or less on the
lines of the powers that I'm

534
00:26:21,870 --> 00:26:23,120
erasing right now.

535
00:26:23,120 --> 00:26:26,630

536
00:26:26,630 --> 00:26:28,360
I want to tell you
about series.

537
00:26:28,360 --> 00:26:30,420
Which are kind of borderline
convergent.

538
00:26:30,420 --> 00:26:33,720
And then next time when we talk
about powers series we'll

539
00:26:33,720 --> 00:26:36,080
come back to this very important
series which is the

540
00:26:36,080 --> 00:26:37,330
most important one.

541
00:26:37,330 --> 00:26:40,680

542
00:26:40,680 --> 00:26:47,400
So now let's talk about some
series general notations.

543
00:26:47,400 --> 00:26:49,970
And this will help you
with the last bit.

544
00:26:49,970 --> 00:26:53,810

545
00:26:53,810 --> 00:26:57,210
This is going to be pretty much
the same as what we did

546
00:26:57,210 --> 00:27:00,890
for improper integrals.

547
00:27:00,890 --> 00:27:04,430
Namely, first of all I'm going
to have capital S N which is

548
00:27:04,430 --> 00:27:10,610
the sum of a n, n equals 0 to
capital N. And this is what

549
00:27:10,610 --> 00:27:12,220
we're calling a partial sum.

550
00:27:12,220 --> 00:27:18,050

551
00:27:18,050 --> 00:27:24,670
And then the full limit which is
capital S, if you like, and

552
00:27:24,670 --> 00:27:30,130
N equals 0 to infinity is just
the limit as N goes to

553
00:27:30,130 --> 00:27:32,010
infinity of the Sn's.

554
00:27:32,010 --> 00:27:36,240

555
00:27:36,240 --> 00:27:37,750
And then we have the
same kind of

556
00:27:37,750 --> 00:27:39,770
notation that we had before.

557
00:27:39,770 --> 00:27:45,250
Which is there are these two
choices which is that if the

558
00:27:45,250 --> 00:27:46,500
limit exists.

559
00:27:46,500 --> 00:27:50,450

560
00:27:50,450 --> 00:27:51,830
That's the green choice.

561
00:27:51,830 --> 00:27:54,460
And we say it converges.

562
00:27:54,460 --> 00:28:00,830
So we say the series
converges.

563
00:28:00,830 --> 00:28:10,900
And then the other case which
is the limit does not exist.

564
00:28:10,900 --> 00:28:12,240
And we can say the
series diverges.

565
00:28:12,240 --> 00:28:20,560

566
00:28:20,560 --> 00:28:22,026
Question.

567
00:28:22,026 --> 00:28:23,276
AUDIENCE: [INAUDIBLE PHRASE]

568
00:28:23,276 --> 00:28:26,480

569
00:28:26,480 --> 00:28:29,290
PROFESSOR: The question was
how did I get to this?

570
00:28:29,290 --> 00:28:31,960
And I will do that next time but
in fact of course you've

571
00:28:31,960 --> 00:28:33,240
seen in high school.

572
00:28:33,240 --> 00:28:34,630
Right this a--

573
00:28:34,630 --> 00:28:35,930
Yeah.

574
00:28:35,930 --> 00:28:36,860
Yeah.

575
00:28:36,860 --> 00:28:40,000
We'll do that next time.

576
00:28:40,000 --> 00:28:42,180
The question was how
did we arrive--

577
00:28:42,180 --> 00:28:43,820
sorry I didn't tell you the
question --the question was

578
00:28:43,820 --> 00:28:45,060
how do we arrive at this
formula on the

579
00:28:45,060 --> 00:28:46,590
right hand side here.

580
00:28:46,590 --> 00:28:48,240
But we'll talk about
that next time.

581
00:28:48,240 --> 00:28:53,060

582
00:28:53,060 --> 00:28:54,020
All right.

583
00:28:54,020 --> 00:29:00,170
So here's the basic definition
and what we're going to

584
00:29:00,170 --> 00:29:02,430
recognize about series.

585
00:29:02,430 --> 00:29:07,690
And I'm going to give you a few
examples and then we'll do

586
00:29:07,690 --> 00:29:08,940
something systematic.

587
00:29:08,940 --> 00:29:12,070

588
00:29:12,070 --> 00:29:13,750
So the first example--

589
00:29:13,750 --> 00:29:16,290
well the first example is
the geometric series.

590
00:29:16,290 --> 00:29:19,780
But the first example that I'm
going to discuss now and in a

591
00:29:19,780 --> 00:29:23,830
little bit of detail is this sum
1 over n squared n equals

592
00:29:23,830 --> 00:29:25,080
1 to infinity.

593
00:29:25,080 --> 00:29:28,580

594
00:29:28,580 --> 00:29:34,050
It turns out that this series
is very analogous--

595
00:29:34,050 --> 00:29:38,640
and we'll develop this analogy
carefully --the integral from

596
00:29:38,640 --> 00:29:41,065
1 to x d x over x squared.

597
00:29:41,065 --> 00:29:46,220
And we're going to develop this
analogy in detail later

598
00:29:46,220 --> 00:29:47,920
in this lecture.

599
00:29:47,920 --> 00:29:50,610
And this one is one of the
ones-- so now you have to go

600
00:29:50,610 --> 00:29:52,840
back and actually remember, this
is one of the ones you

601
00:29:52,840 --> 00:29:54,510
really want to memorize.

602
00:29:54,510 --> 00:29:56,720
And you should especially pay
attention to the ones with an

603
00:29:56,720 --> 00:29:58,570
infinity in them.

604
00:29:58,570 --> 00:29:59,820
This one is convergent.

605
00:29:59,820 --> 00:30:03,270

606
00:30:03,270 --> 00:30:04,520
And this series is convergent.

607
00:30:04,520 --> 00:30:11,070
Now it turns out that evaluating
this is very easy.

608
00:30:11,070 --> 00:30:12,820
This is 1.

609
00:30:12,820 --> 00:30:15,310
It's easy to calculate.

610
00:30:15,310 --> 00:30:19,500
Evaluating this is
very tricky.

611
00:30:19,500 --> 00:30:21,610
And Euler did it.

612
00:30:21,610 --> 00:30:26,440
And the answer is pi
squared over 6.

613
00:30:26,440 --> 00:30:29,050
That's an amazing calculation.

614
00:30:29,050 --> 00:30:33,570
And it was done very early in
the history of mathematics.

615
00:30:33,570 --> 00:30:36,450
If you look at another
example--

616
00:30:36,450 --> 00:30:42,300
so maybe example two here -if
you look at 1 over n cubed.

617
00:30:42,300 --> 00:30:46,580
n equals-- well you can't start
here at 0 by the way.

618
00:30:46,580 --> 00:30:48,930
I get to start wherever I
want in these series.

619
00:30:48,930 --> 00:30:49,920
Here I start with 0.

620
00:30:49,920 --> 00:30:51,220
Here I started with 1.

621
00:30:51,220 --> 00:30:54,030
And notice the reason why I I
started, it was a bad idea to

622
00:30:54,030 --> 00:30:58,130
start with 0 was that 1
over 0 is undefined.

623
00:30:58,130 --> 00:30:58,360
Right?

624
00:30:58,360 --> 00:31:00,430
So I'm just starting where
it's convenient for me.

625
00:31:00,430 --> 00:31:03,800
And since I'm interested mostly
in the tale behavior it

626
00:31:03,800 --> 00:31:06,150
doesn't matter to me so
much where I start.

627
00:31:06,150 --> 00:31:08,890
Although if I want an exact
answer I need to start exactly

628
00:31:08,890 --> 00:31:10,165
at n equals 1.

629
00:31:10,165 --> 00:31:10,890
All right.

630
00:31:10,890 --> 00:31:17,932
This one is similar to
this integral here.

631
00:31:17,932 --> 00:31:18,700
All right.

632
00:31:18,700 --> 00:31:20,440
Which is convergent again.

633
00:31:20,440 --> 00:31:22,390
So there's a number
that you get.

634
00:31:22,390 --> 00:31:26,840
And let's see what is it
something like two thirds or

635
00:31:26,840 --> 00:31:28,470
something like that,
all right, for

636
00:31:28,470 --> 00:31:30,190
this for this number.

637
00:31:30,190 --> 00:31:32,440
Or a third.

638
00:31:32,440 --> 00:31:33,110
What is it?

639
00:31:33,110 --> 00:31:33,690
No a half.

640
00:31:33,690 --> 00:31:35,000
I guess it's a half.

641
00:31:35,000 --> 00:31:37,110
This one is a half.

642
00:31:37,110 --> 00:31:39,430
You check that I'm not positive,
but anyway just

643
00:31:39,430 --> 00:31:42,150
doing it in my head quickly
it seems to be a half.

644
00:31:42,150 --> 00:31:44,100
Anyway it's an easy number
to calculate.

645
00:31:44,100 --> 00:31:48,940
This one over here stumped
mathematicians

646
00:31:48,940 --> 00:31:51,870
basically for all time.

647
00:31:51,870 --> 00:31:56,250
It doesn't have any kind of
elementary form like this.

648
00:31:56,250 --> 00:31:59,910
And it was only very recently
proved to be rational.

649
00:31:59,910 --> 00:32:02,110
People couldn't even couldn't
even decide whether this was a

650
00:32:02,110 --> 00:32:05,260
rational number or not.

651
00:32:05,260 --> 00:32:07,050
But anyway that's been resolved
it is an irrational

652
00:32:07,050 --> 00:32:09,540
number which is what
people suspected.

653
00:32:09,540 --> 00:32:10,650
Yeah question.

654
00:32:10,650 --> 00:32:11,900
AUDIENCE: [INAUDIBLE PHRASE]

655
00:32:11,900 --> 00:32:14,360

656
00:32:14,360 --> 00:32:16,350
PROFESSOR: Yeah sorry.

657
00:32:16,350 --> 00:32:17,600
OK.

658
00:32:17,600 --> 00:32:19,820

659
00:32:19,820 --> 00:32:25,070
I violated a rule
of mathematics--

660
00:32:25,070 --> 00:32:26,720
you said why is this similar?

661
00:32:26,720 --> 00:32:29,140
I thought that similar
was something else.

662
00:32:29,140 --> 00:32:30,380
And you're absolutely right.

663
00:32:30,380 --> 00:32:33,520
And I violated a rule
of mathematics.

664
00:32:33,520 --> 00:32:36,930
Which is that I used
this symbol for

665
00:32:36,930 --> 00:32:38,180
two different things.

666
00:32:38,180 --> 00:32:41,210

667
00:32:41,210 --> 00:32:42,820
I should have written
this symbol here.

668
00:32:42,820 --> 00:32:43,530
All right.

669
00:32:43,530 --> 00:32:45,260
I'll create a new symbol here.

670
00:32:45,260 --> 00:32:48,920
The question of whether this
converges or this converges.

671
00:32:48,920 --> 00:32:51,920
These are the the same
type of question.

672
00:32:51,920 --> 00:32:53,380
And we'll see why
they're the same

673
00:32:53,380 --> 00:32:55,000
question it in a few minutes.

674
00:32:55,000 --> 00:32:59,780
But in fact the wiggle I used
similar I used for the

675
00:32:59,780 --> 00:33:02,360
connection between functions.

676
00:33:02,360 --> 00:33:05,240
The things that are really
similar are that 1 over n

677
00:33:05,240 --> 00:33:10,230
resembles 1 over x squared.

678
00:33:10,230 --> 00:33:12,790
So I apologize I didn't--

679
00:33:12,790 --> 00:33:14,040
AUDIENCE: [INAUDIBLE PHRASE]

680
00:33:14,040 --> 00:33:15,890

681
00:33:15,890 --> 00:33:18,140
PROFESSOR: Oh you thought
that this was the

682
00:33:18,140 --> 00:33:19,290
definition of that.

683
00:33:19,290 --> 00:33:20,950
That's actually the reason
why these things

684
00:33:20,950 --> 00:33:21,990
correspond so closely.

685
00:33:21,990 --> 00:33:25,560
That is that the Riemann
sum is close to this.

686
00:33:25,560 --> 00:33:27,520
But that doesn't mean
they're equal.

687
00:33:27,520 --> 00:33:31,950
The Riemann sum only works when
the delta x goes to 0.

688
00:33:31,950 --> 00:33:33,700
The way that we're going to get
a connection between these

689
00:33:33,700 --> 00:33:38,490
two, as we will just a second,
is with a Riemann sum with.

690
00:33:38,490 --> 00:33:44,780
What we're going to use is
a Riemann's sum with

691
00:33:44,780 --> 00:33:47,180
delta x equals 1.

692
00:33:47,180 --> 00:33:49,617
All right and then that will
be the connection between.

693
00:33:49,617 --> 00:33:53,480
All right that's absolutely
right.

694
00:33:53,480 --> 00:33:54,730
All right.

695
00:33:54,730 --> 00:33:57,020

696
00:33:57,020 --> 00:34:00,520
So in order to illustrate
exactly this idea that you've

697
00:34:00,520 --> 00:34:01,520
just come up with.

698
00:34:01,520 --> 00:34:03,110
And in fact that we're
going to use.

699
00:34:03,110 --> 00:34:06,150
We'll do the same thing but
we're going to do it on the

700
00:34:06,150 --> 00:34:08,000
example sum 1 over n.

701
00:34:08,000 --> 00:34:13,380

702
00:34:13,380 --> 00:34:20,630
So here's example 3 and it's
going to be sum 1 over n, n

703
00:34:20,630 --> 00:34:21,880
equals 1 to infinity.

704
00:34:21,880 --> 00:34:24,090

705
00:34:24,090 --> 00:34:28,420
And what we're now going to see
is that it corresponds to

706
00:34:28,420 --> 00:34:29,670
this integral here.

707
00:34:29,670 --> 00:34:32,620

708
00:34:32,620 --> 00:34:34,850
And we're going to
show therefore

709
00:34:34,850 --> 00:34:37,870
that this thing diverges.

710
00:34:37,870 --> 00:34:40,300
But we're going to do
this more carefully.

711
00:34:40,300 --> 00:34:43,444
We're going to do this in some
detail so that you see what it

712
00:34:43,444 --> 00:34:47,390
is, that the correspondence is
between these quantities.

713
00:34:47,390 --> 00:34:49,890
And the same sort of
reasoning applies

714
00:34:49,890 --> 00:34:51,275
to these other examples.

715
00:34:51,275 --> 00:34:55,820

716
00:34:55,820 --> 00:34:59,180
So here we go.

717
00:34:59,180 --> 00:35:05,600
I'm going to take the integral
and draw the picture of the

718
00:35:05,600 --> 00:35:07,220
Riemann's sum.

719
00:35:07,220 --> 00:35:11,200
So here's the level one and
here's the function y

720
00:35:11,200 --> 00:35:13,840
equals 1 over x.

721
00:35:13,840 --> 00:35:15,730
And I'm going to take
the Riemann's sum.

722
00:35:15,730 --> 00:35:21,980

723
00:35:21,980 --> 00:35:25,200
With delta x equals 1.

724
00:35:25,200 --> 00:35:29,140
And that's going to be closely
connected to the series that I

725
00:35:29,140 --> 00:35:35,510
have. But now I have to decide
whether I want a lower

726
00:35:35,510 --> 00:35:37,820
Riemann's sum or an upper
Riemann's sum.

727
00:35:37,820 --> 00:35:39,920
And actually I'm going to check
both of them because

728
00:35:39,920 --> 00:35:41,170
both of them are illuminating.

729
00:35:41,170 --> 00:35:44,620

730
00:35:44,620 --> 00:35:47,000
First we'll do the upper
Riemann's sum.

731
00:35:47,000 --> 00:35:48,700
Now that's this staircase
here.

732
00:35:48,700 --> 00:35:51,780

733
00:35:51,780 --> 00:35:54,600
So we'll call this the
upper Riemann's sum.

734
00:35:54,600 --> 00:35:58,130

735
00:35:58,130 --> 00:35:59,710
And let's check what
it's levels are.

736
00:35:59,710 --> 00:36:01,560
This is not to scale.

737
00:36:01,560 --> 00:36:03,250
This level should be a half.

738
00:36:03,250 --> 00:36:05,610
So if this is 1 and this is 2
and that level was supposed to

739
00:36:05,610 --> 00:36:10,340
be a half and this next level
should be a third.

740
00:36:10,340 --> 00:36:12,830
That's how the Riemann's
sums are working out.

741
00:36:12,830 --> 00:36:17,040

742
00:36:17,040 --> 00:36:21,770
And now I have the following
phenomenon.

743
00:36:21,770 --> 00:36:24,640
Let's cut it off at
the nth stage.

744
00:36:24,640 --> 00:36:27,830
So that means that I'm going,
the integral is from 1

745
00:36:27,830 --> 00:36:30,760
to n d x over x.

746
00:36:30,760 --> 00:36:32,780
And the Riemann's
sum is something

747
00:36:32,780 --> 00:36:33,690
that's bigger than it.

748
00:36:33,690 --> 00:36:37,820
Because the areas are enclosing
the area of the

749
00:36:37,820 --> 00:36:40,300
curved region.

750
00:36:40,300 --> 00:36:43,610
And that's going to be the area
of the first box which is

751
00:36:43,610 --> 00:36:50,826
1, plus the area of the second
box which is a half plus the

752
00:36:50,826 --> 00:36:54,150
area of the third box
which is a third.

753
00:36:54,150 --> 00:36:58,940
All the way up the last
one but the last one

754
00:36:58,940 --> 00:37:00,840
starts at n minus 1.

755
00:37:00,840 --> 00:37:03,450
So it has 1 over n minus 1.

756
00:37:03,450 --> 00:37:05,780
There are not n boxes here.

757
00:37:05,780 --> 00:37:07,960
They're only n minus 1 boxes.

758
00:37:07,960 --> 00:37:11,830
Because the distance between
1 and n is n minus one.

759
00:37:11,830 --> 00:37:17,330
Right so this is n
minus 1 terms.

760
00:37:17,330 --> 00:37:25,140
However, if I use the notation
for partial sum.

761
00:37:25,140 --> 00:37:29,920
Which is 1 plus 1 over 2 plus
all the way up to 1 over n

762
00:37:29,920 --> 00:37:33,530
minus 1 plus 1 over n.

763
00:37:33,530 --> 00:37:35,750
In other words I go out to the
nth one which is what I would

764
00:37:35,750 --> 00:37:37,370
ordinarily do.

765
00:37:37,370 --> 00:37:42,900
Then this sum that I have here
certainly is less than Sn.

766
00:37:42,900 --> 00:37:47,850
Because there's one
more term there.

767
00:37:47,850 --> 00:37:51,070
And so here I have an
integral which is

768
00:37:51,070 --> 00:37:53,570
underneath this sum Sn.

769
00:37:53,570 --> 00:38:01,700

770
00:38:01,700 --> 00:38:21,000
Now this is going to allow us
to prove conclusively that

771
00:38:21,000 --> 00:38:21,156
the-- so I'm just going to
rewrite this-- prove

772
00:38:21,156 --> 00:38:22,350
conclusively that the
sum diverges.

773
00:38:22,350 --> 00:38:23,340
Why is that?

774
00:38:23,340 --> 00:38:26,000
Because this term here
we can calculate.

775
00:38:26,000 --> 00:38:29,470
This is the log x evaluated
at 1 and n.

776
00:38:29,470 --> 00:38:34,850
Which is the same thing
as log n minus 0.

777
00:38:34,850 --> 00:38:39,270
All right, the quantity log
n minus log 1 which is 0.

778
00:38:39,270 --> 00:38:46,892
And so what we have here is that
log n is less than S n.

779
00:38:46,892 --> 00:38:51,455
All right and clearly this
goes to infinity right.

780
00:38:51,455 --> 00:38:57,830
As n goes to infinity this
thing goes to infinity.

781
00:38:57,830 --> 00:38:58,435
So we're done.

782
00:38:58,435 --> 00:39:00,330
All right we've shown
divergence.

783
00:39:00,330 --> 00:39:08,730

784
00:39:08,730 --> 00:39:16,000
Now the way I'm going to use the
lower Riemann's sum is to

785
00:39:16,000 --> 00:39:21,960
recognize that we've captured
the rate appropriately.

786
00:39:21,960 --> 00:39:24,560
That is not only do I have a
lower bound like this but I

787
00:39:24,560 --> 00:39:27,860
have an upper bound which
is very similar.

788
00:39:27,860 --> 00:39:29,170
So if I use the upper--

789
00:39:29,170 --> 00:39:38,986
Riemann's oh sorry --the lower
Riemann's sum again with delta

790
00:39:38,986 --> 00:39:40,236
x equals 1.

791
00:39:40,236 --> 00:39:43,760

792
00:39:43,760 --> 00:39:55,530
Then I have that the integral
from 1 to n of d x over x is

793
00:39:55,530 --> 00:39:56,840
bigger than--

794
00:39:56,840 --> 00:39:58,760
Well what are the terms going
to be if fit sit them

795
00:39:58,760 --> 00:40:00,640
underneath?

796
00:40:00,640 --> 00:40:03,210
If I fit them underneath I'm
missing the first term.

797
00:40:03,210 --> 00:40:05,600
That is the box is going
to be half height.

798
00:40:05,600 --> 00:40:08,300
It's going to be this
lower piece.

799
00:40:08,300 --> 00:40:10,480
So I'm missing this
first term.

800
00:40:10,480 --> 00:40:17,240
So it'll be a half plus a third
plus all right, it will

801
00:40:17,240 --> 00:40:18,150
keep on going.

802
00:40:18,150 --> 00:40:22,150
But now the last one instead
of being 1 over n minus 1.

803
00:40:22,150 --> 00:40:23,840
It's going to be 1 over n.

804
00:40:23,840 --> 00:40:27,540
This is again a total of the n
minus 1 terms. This is the

805
00:40:27,540 --> 00:40:28,790
lower Riemann's sum.

806
00:40:28,790 --> 00:40:31,190

807
00:40:31,190 --> 00:40:40,232
And now we can recognize that
this is exactly equal to--

808
00:40:40,232 --> 00:40:40,574
well so I'll put it over here--
this is exactly equal

809
00:40:40,574 --> 00:40:43,520
to S n minus 1 minus
the first term.

810
00:40:43,520 --> 00:40:45,260
So we missed the first
term but we got

811
00:40:45,260 --> 00:40:47,110
all the rest of them.

812
00:40:47,110 --> 00:40:52,432
So if I put this to the other
side remember this is log n.

813
00:40:52,432 --> 00:40:53,770
All right.

814
00:40:53,770 --> 00:40:56,170
If I put this to the other side
I have the other side of

815
00:40:56,170 --> 00:40:57,130
this bound.

816
00:40:57,130 --> 00:41:07,010
I have that S n is less than if
I reverse it log n plus 1.

817
00:41:07,010 --> 00:41:09,040
And so I've trapped it
on the other side.

818
00:41:09,040 --> 00:41:10,950
And here I have the
lower bound.

819
00:41:10,950 --> 00:41:13,170
So I'm going to combine
those together.

820
00:41:13,170 --> 00:41:18,030
So all hold I have this
correspondence here.

821
00:41:18,030 --> 00:41:23,150
It is the size of a log n is
trapped between the-- sorry--

822
00:41:23,150 --> 00:41:26,150
the size of S n which is
relatively hard to calculate

823
00:41:26,150 --> 00:41:30,780
and understand exactly is
trapped between log n

824
00:41:30,780 --> 00:41:34,260
and log n plus 1.

825
00:41:34,260 --> 00:41:35,160
Yeah question.

826
00:41:35,160 --> 00:41:36,410
AUDIENCE: [INAUDIBLE PHRASE]

827
00:41:36,410 --> 00:41:46,280

828
00:41:46,280 --> 00:41:47,670
PROFESSOR: This step here
is the step that

829
00:41:47,670 --> 00:41:49,550
you're concerned about.

830
00:41:49,550 --> 00:41:54,580
So this step is a geometric
argument which is

831
00:41:54,580 --> 00:41:57,070
analogous to this step.

832
00:41:57,070 --> 00:42:01,320
All right it's the same
type of argument.

833
00:42:01,320 --> 00:42:05,100
And in this case it's that the
rectangle's are on top and so

834
00:42:05,100 --> 00:42:07,820
the area represented on the
right hand side is less than

835
00:42:07,820 --> 00:42:09,840
the area represented
on this side.

836
00:42:09,840 --> 00:42:12,130
And this is the same type
of thing except that the

837
00:42:12,130 --> 00:42:14,400
rectangle's are underneath.

838
00:42:14,400 --> 00:42:18,150
So the sum of the areas of the
rectangle is less than the

839
00:42:18,150 --> 00:42:19,400
area under the curve.

840
00:42:19,400 --> 00:42:23,840

841
00:42:23,840 --> 00:42:24,240
All right.

842
00:42:24,240 --> 00:42:27,380
So I've now trapped
this quantity.

843
00:42:27,380 --> 00:42:34,440
And I'm now going to state the
sort of general results.

844
00:42:34,440 --> 00:42:38,420

845
00:42:38,420 --> 00:42:42,230
So here's what's known as
integral comparison.

846
00:42:42,230 --> 00:42:45,710
It's this double arrow
correspondence in the general

847
00:42:45,710 --> 00:42:54,580
case, for a very general case.

848
00:42:54,580 --> 00:42:57,130
There are actually even more
cases where it works.

849
00:42:57,130 --> 00:43:01,510
But this is a good case
and convenient.

850
00:43:01,510 --> 00:43:02,845
Now this is called integral
comparison.

851
00:43:02,845 --> 00:43:07,060

852
00:43:07,060 --> 00:43:11,720
And it comes with hypotheses
but it follows the same

853
00:43:11,720 --> 00:43:22,440
argument that I just gave. If f
of x is decreasing and it's

854
00:43:22,440 --> 00:43:38,150
positive then the sum f n, n
equals 1 to infinity minus the

855
00:43:38,150 --> 00:43:44,740
integral from 1 to infinity
of f of x d x is

856
00:43:44,740 --> 00:43:45,990
less than f of 1.

857
00:43:45,990 --> 00:43:50,060

858
00:43:50,060 --> 00:43:51,590
That's basically
what we showed.

859
00:43:51,590 --> 00:43:54,140
We showed that the difference
between s n and log

860
00:43:54,140 --> 00:43:55,930
n was at most 1.

861
00:43:55,930 --> 00:43:59,660

862
00:43:59,660 --> 00:43:59,820
All right.

863
00:43:59,820 --> 00:44:17,730
Now if both of them are, and
the sum and the integral

864
00:44:17,730 --> 00:44:25,380
converge or diverge together.

865
00:44:25,380 --> 00:44:27,750
That is they either both
converge or both diverge.

866
00:44:27,750 --> 00:44:30,570
This is the type of test that
we like because then we can

867
00:44:30,570 --> 00:44:33,860
just convert the question of
convergence over here to this

868
00:44:33,860 --> 00:44:37,770
question of convergence over
on the other side.

869
00:44:37,770 --> 00:44:42,430
Now I remind you that it's
incredibly hard to calculate

870
00:44:42,430 --> 00:44:44,870
these numbers.

871
00:44:44,870 --> 00:44:47,220
Whereas these numbers are
easier to calculate.

872
00:44:47,220 --> 00:44:50,390
Our goal is to reduce things
to simpler things.

873
00:44:50,390 --> 00:44:53,960
And in this case sums, infinite
sums are much harder

874
00:44:53,960 --> 00:44:55,210
than infinite integrals.

875
00:44:55,210 --> 00:45:00,080

876
00:45:00,080 --> 00:45:03,150
All right so that's the
integral comparison.

877
00:45:03,150 --> 00:45:13,030
And now I have one last bit on
comparisons that I need to

878
00:45:13,030 --> 00:45:13,840
tell you about.

879
00:45:13,840 --> 00:45:16,310
And this is very much like what
we did with integrals.

880
00:45:16,310 --> 00:45:18,240
Which is a so called
limit comparison.

881
00:45:18,240 --> 00:45:29,200

882
00:45:29,200 --> 00:45:36,950
The limit comparison says the
following if f of n is

883
00:45:36,950 --> 00:45:38,270
similar to g of n.

884
00:45:38,270 --> 00:45:46,880
You will recall that means f of
n over g of n tends to 1 as

885
00:45:46,880 --> 00:45:48,130
n goes to infinity.

886
00:45:48,130 --> 00:45:51,760

887
00:45:51,760 --> 00:45:55,000
And we're in the
positive case.

888
00:45:55,000 --> 00:45:57,535
So let's just say
g n is positive.

889
00:45:57,535 --> 00:46:03,920

890
00:46:03,920 --> 00:46:11,350
Then-- that doesn't even, well--
then sum f n sum g n

891
00:46:11,350 --> 00:46:20,460
either both, same thing as
above, either both converge or

892
00:46:20,460 --> 00:46:21,884
both diverge.

893
00:46:21,884 --> 00:46:27,488

894
00:46:27,488 --> 00:46:28,770
All right.

895
00:46:28,770 --> 00:46:31,240
This is just saying that if they
behave the same way in

896
00:46:31,240 --> 00:46:36,320
the tail, which is all we really
care about, then they

897
00:46:36,320 --> 00:46:40,820
have similar behavior, similar
convergence properties.

898
00:46:40,820 --> 00:46:44,620

899
00:46:44,620 --> 00:46:45,870
And let me give you
a couple examples.

900
00:46:45,870 --> 00:46:50,370

901
00:46:50,370 --> 00:46:55,850
So here's one example if you
take the sum 1 over n squared

902
00:46:55,850 --> 00:46:57,100
plus 1 square root.

903
00:46:57,100 --> 00:47:01,890

904
00:47:01,890 --> 00:47:05,350
This is going to be replaced
by something simpler.

905
00:47:05,350 --> 00:47:07,240
Which is the main term here.

906
00:47:07,240 --> 00:47:12,370
Which is 1 over square root of
n squared which we recognize

907
00:47:12,370 --> 00:47:15,100
as sum 1 over n which
diverges.

908
00:47:15,100 --> 00:47:17,920

909
00:47:17,920 --> 00:47:20,440
So this guy is one
of the red guys.

910
00:47:20,440 --> 00:47:24,300

911
00:47:24,300 --> 00:47:26,410
On the red team.

912
00:47:26,410 --> 00:47:30,330
Now we have another example.

913
00:47:30,330 --> 00:47:33,370

914
00:47:33,370 --> 00:47:39,180
Which is let's say the square
root of n, I don't know, to

915
00:47:39,180 --> 00:47:43,080
the fifth minus n squared.

916
00:47:43,080 --> 00:47:45,710
Now if you have something where
it's negative in the

917
00:47:45,710 --> 00:47:48,400
denominator you kind of do
have to watch out that

918
00:47:48,400 --> 00:47:49,540
denominator makes sense.

919
00:47:49,540 --> 00:47:50,570
It isn't 0.

920
00:47:50,570 --> 00:47:53,042
So we're going to be careful and
start this at n equals 2.

921
00:47:53,042 --> 00:48:00,570
In which case, the first term,
I don't like 1 over 0 as a

922
00:48:00,570 --> 00:48:01,720
term in my series.

923
00:48:01,720 --> 00:48:04,380
So I'm just going to be a little
careful about how-- as

924
00:48:04,380 --> 00:48:05,710
I said I was kind
of lazy here.

925
00:48:05,710 --> 00:48:09,930
I could have started this
one at 0 for instance.

926
00:48:09,930 --> 00:48:10,810
All right.

927
00:48:10,810 --> 00:48:14,110
So here's the picture.

928
00:48:14,110 --> 00:48:18,240
Now this I just replace by it's
main term which is 1 over

929
00:48:18,240 --> 00:48:20,340
n to the fifth square root.

930
00:48:20,340 --> 00:48:25,975
Which is sum 1 over n to the
five halves which converges.

931
00:48:25,975 --> 00:48:28,705

932
00:48:28,705 --> 00:48:29,520
All right.

933
00:48:29,520 --> 00:48:30,800
The power is bigger than 1.

934
00:48:30,800 --> 00:48:33,960
1 is the divider for these
things and it just misses.

935
00:48:33,960 --> 00:48:38,752
This one converges.

936
00:48:38,752 --> 00:48:45,620
All right so these are the
typical ways in which these

937
00:48:45,620 --> 00:48:47,355
convergence processes
are used.

938
00:48:47,355 --> 00:48:47,870
All right.

939
00:48:47,870 --> 00:48:49,930
So I have one more
thing for you.

940
00:48:49,930 --> 00:48:52,600
Which is an advertisement
for next time.

941
00:48:52,600 --> 00:48:56,360
And I have this demo here
which I will grab.

942
00:48:56,360 --> 00:48:58,300
But you will see
this next time.

943
00:48:58,300 --> 00:49:01,120
So here's a question for you to
think about overnight but

944
00:49:01,120 --> 00:49:04,240
don't ask friends you have to
think about it yourself.

945
00:49:04,240 --> 00:49:05,492
So here's the problem.

946
00:49:05,492 --> 00:49:08,180
Here are some blocks which
I acquired when

947
00:49:08,180 --> 00:49:09,430
my kids left home.

948
00:49:09,430 --> 00:49:12,190

949
00:49:12,190 --> 00:49:20,550
Anyway yeah that'll happen to
you too in about four years.

950
00:49:20,550 --> 00:49:26,100
So now here you are,
these are blocks.

951
00:49:26,100 --> 00:49:27,930
So now here's the question
that we're going to

952
00:49:27,930 --> 00:49:29,860
deal with next time.

953
00:49:29,860 --> 00:49:32,250
I'm going to build it, maybe
I'll put it over here because

954
00:49:32,250 --> 00:49:34,890
I want to have some room
to head this way.

955
00:49:34,890 --> 00:49:38,990
I want to stack them
up so that--

956
00:49:38,990 --> 00:49:43,430
oh didn't work --going
to stack them up in

957
00:49:43,430 --> 00:49:44,430
the following way.

958
00:49:44,430 --> 00:49:48,530
I want to do it so that the top
one is completely to the

959
00:49:48,530 --> 00:49:51,540
right of the bottom one.

960
00:49:51,540 --> 00:49:53,170
That's the question
can I do that?

961
00:49:53,170 --> 00:49:55,840
Can I get?

962
00:49:55,840 --> 00:49:57,870
Can I build this up?

963
00:49:57,870 --> 00:50:01,980
So what's let's see here.

964
00:50:01,980 --> 00:50:03,140
I just seem to be missing--

965
00:50:03,140 --> 00:50:05,310
but anyway what I'm going to
do is I'm going to try to

966
00:50:05,310 --> 00:50:08,460
build this and we're going to
see how far we can get with

967
00:50:08,460 --> 00:50:10,880
this next time.

968
00:50:10,880 --> 00:50:11,678