1
00:00:00,000 --> 00:00:02,330
The following content is
provided under a Creative

2
00:00:02,330 --> 00:00:03,144
Commons license.

3
00:00:03,144 --> 00:00:05,460
Your support will help
MIT OpenCourseWare

4
00:00:05,460 --> 00:00:09,646
continue to offer high quality
educational resources for free.

5
00:00:09,646 --> 00:00:12,260
To make a donation or to
view additional materials

6
00:00:12,260 --> 00:00:16,150
from hundreds of MIT courses
visit MIT OpenCourseWare

7
00:00:16,150 --> 00:00:17,090
at ocw.mit.edu.

8
00:00:17,090 --> 00:00:21,344


9
00:00:21,344 --> 00:00:22,760
PROFESSOR: Today
we are continuing

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

11
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


13
00:00:30,080 --> 00:00:33,510
What we were discussing at
the very end of last time

14
00:00:33,510 --> 00:00:36,195
was improper integrals.

15
00:00:36,195 --> 00:00:41,560


16
00:00:41,560 --> 00:00:44,570
Now and these are going
to be improper integrals

17
00:00:44,570 --> 00:00:48,220
of the second kind.

18
00:00:48,220 --> 00:00:50,990
By second kind I mean that
they have a singularity

19
00:00:50,990 --> 00:00:52,095
at a finite place.

20
00:00:52,095 --> 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.

23
00:01:00,140 --> 00:01:03,380
Same sort of thing as we
did when the singularity

24
00:01:03,380 --> 00:01:04,120
was at infinity.

25
00:01:04,120 --> 00:01:09,440
So if you have the integral
from 0 to 1 of f(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 --> 00:01:17,780
as a goes to 0 from above,
the integral from a to 1

28
00:01:17,780 --> 00:01:21,150
of f(x) dx.

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 --> 00:01:35,180
And we don't know whether
the area enclosed is

34
00:01:35,180 --> 00:01:39,040
going to be infinite or
finite and so we cut it off

35
00:01:39,040 --> 00:01:40,680
at some place a.

36
00:01:40,680 --> 00:01:44,220
And we let a go to 0 from above.

37
00:01:44,220 --> 00:01:46,800
So really it's 0+.

38
00:01:46,800 --> 00:01:49,820
So we're coming in
from the right here.

39
00:01:49,820 --> 00:01:52,810
And we're counting up
the area in this chunk.

40
00:01:52,810 --> 00:01:56,520
And we're seeing as it expands
whether it goes to infinity

41
00:01:56,520 --> 00:01:59,290
or whether it tends
to some finite limit.

42
00:01:59,290 --> 00:02:02,990
Right, so this is the example
and this is the definition.

43
00:02:02,990 --> 00:02:07,000
And just as we did for the
other kind of improper integral,

44
00:02:07,000 --> 00:02:13,747
we say that this converges --
so that's the key word here --

45
00:02:13,747 --> 00:02:24,220
if the limit is finite,
exists maybe I should just say

46
00:02:24,220 --> 00:02:30,773
and diverges if not.

47
00:02:30,773 --> 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 --> 00:02:55,402


54
00:02:55,402 --> 00:02:57,360
And this just, you almost
don't notice the fact

55
00:02:57,360 --> 00:02:59,920
that it 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 --> 00:03:05,860
But if you evaluate it -- first
of all we always write this

58
00:03:05,860 --> 00:03:06,860
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.

62
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^(1/2) times 2.

64
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.

66
00:03:23,750 --> 00:03:25,080
And I get 2.

67
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 --> 00:03:42,140
The second example,
being not systematic

72
00:03:42,140 --> 00:03:44,300
but really giving you
the principal examples

73
00:03:44,300 --> 00:03:51,650
that we'll be thinking about,
is this one here, dx / x.

74
00:03:51,650 --> 00:03:54,200
And this one gives
you the antiderivative

75
00:03:54,200 --> 00:03:56,070
as the logarithm.

76
00:03:56,070 --> 00:03:58,180
Evaluated at 0 and 1.

77
00:03:58,180 --> 00:04:00,190
And now again you have
to have this thought

78
00:04:00,190 --> 00:04:02,770
process in your mind that
you're really taking the limit.

79
00:04:02,770 --> 00:04:06,440
But this is going to be the
log of 1 minus the log of 0.

80
00:04:06,440 --> 00:04:07,830
Really the log of 0 from above.

81
00:04:07,830 --> 00:04:10,970
There is no such thing as
the log of 0 from below.

82
00:04:10,970 --> 00:04:12,820
And this is minus infinity.

83
00:04:12,820 --> 00:04:19,020
So it's 0 minus minus infinity,
which is plus infinity.

84
00:04:19,020 --> 00:04:20,213
And so this one diverges.

85
00:04:20,213 --> 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 --> 00:04:45,220
if you work out this thing
for dx / x^p from 0 to 1.

89
00:04:45,220 --> 00:04:51,820
What you're going to find is
that it's 1/(1-p) when p is

90
00:04:51,820 --> 00:04:53,020
less than 1.

91
00:04:53,020 --> 00:05:00,240
And it diverges for p >= 1.

92
00:05:00,240 --> 00:05:02,750


93
00:05:02,750 --> 00:05:07,880
Now that's the final result. If
you carry out this integration

94
00:05:07,880 --> 00:05:10,920
it's not difficult.

95
00:05:10,920 --> 00:05:14,850
All right so now
I just want to try

96
00:05:14,850 --> 00:05:17,270
to help you to remember this.

97
00:05:17,270 --> 00:05:20,500
And to think about how
you should think about it.

98
00:05:20,500 --> 00:05:23,890
So I'm going to say
it in a few more ways.

99
00:05:23,890 --> 00:05:26,850
All right just repeat
what I've said already

100
00:05:26,850 --> 00:05:32,020
but try to get it to
percolate and absorb itself.

101
00:05:32,020 --> 00:05:34,270
And in order to do
that I have to make

102
00:05:34,270 --> 00:05:37,750
the contrast between the
kind of improper integral

103
00:05:37,750 --> 00:05:39,410
that I was dealing with before.

104
00:05:39,410 --> 00:05:43,720
Which was not as x goes to 0
here but as x goes to infinity,

105
00:05:43,720 --> 00:05:45,920
the other side.

106
00:05:45,920 --> 00:05:47,023
Let's make this contrast.

107
00:05:47,023 --> 00:05:52,470


108
00:05:52,470 --> 00:05:55,610
First of all, if I
look at the angle

109
00:05:55,610 --> 00:05:57,610
that we have been paying
attention to right now.

110
00:05:57,610 --> 00:06:00,580
We've just considered
things like this.

111
00:06:00,580 --> 00:06:02,520
1 over x to the 1/2.

112
00:06:02,520 --> 00:06:06,600
Which is a lot smaller than 1/x.

113
00:06:06,600 --> 00:06:10,740
Which is a lot smaller
than say 1/x^2.

114
00:06:10,740 --> 00:06:12,050
Which would be another example.

115
00:06:12,050 --> 00:06:14,560
This is as x goes to 0.

116
00:06:14,560 --> 00:06:19,990


117
00:06:19,990 --> 00:06:21,935
So this one's the smallest one.

118
00:06:21,935 --> 00:06:23,310
This one's the
next smallest one.

119
00:06:23,310 --> 00:06:26,820
And this one is very large.

120
00:06:26,820 --> 00:06:30,985
On the other hand it goes
the other way at infinity.

121
00:06:30,985 --> 00:06:36,670


122
00:06:36,670 --> 00:06:39,936
As x tends to infinity.

123
00:06:39,936 --> 00:06:43,140
All right so try to
keep that in mind.

124
00:06:43,140 --> 00:06:48,760
And now I'm going to put a
little box around the bad guys

125
00:06:48,760 --> 00:06:50,100
here.

126
00:06:50,100 --> 00:06:54,710
This one is divergent.

127
00:06:54,710 --> 00:06:57,870
And this one is divergent.

128
00:06:57,870 --> 00:06:59,950
And this one is divergent.

129
00:06:59,950 --> 00:07:01,610
And this one is divergent.

130
00:07:01,610 --> 00:07:03,510
The crossover point is 1/x.

131
00:07:03,510 --> 00:07:05,680
When we get smaller
than that, we

132
00:07:05,680 --> 00:07:07,390
get to things which
are convergent.

133
00:07:07,390 --> 00:07:10,590
When we get smaller than
it on this other scale,

134
00:07:10,590 --> 00:07:12,412
it's convergent.

135
00:07:12,412 --> 00:07:13,995
All right so these
guys are divergent.

136
00:07:13,995 --> 00:07:20,300


137
00:07:20,300 --> 00:07:23,430
So they're associated
with divergent integrals.

138
00:07:23,430 --> 00:07:25,390
The functions
themselves are just

139
00:07:25,390 --> 00:07:27,585
tending towards-- well
these tend to infinity,

140
00:07:27,585 --> 00:07:29,310
and these tend toward 0.

141
00:07:29,310 --> 00:07:33,830
So I'm not talking about
the functions themselves

142
00:07:33,830 --> 00:07:35,190
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 --> 00:07:43,757


145
00:07:43,757 --> 00:07:44,840
I want to draw this again.

146
00:07:44,840 --> 00:07:48,484


147
00:07:48,484 --> 00:07:50,150
And, so I'm just going
to draw a picture

148
00:07:50,150 --> 00:07:51,760
of what it 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 --> 00:08:01,330
So here's the picture
for example of y = 1/x.

151
00:08:01,330 --> 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 y = 1/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 two halves.

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

158
00:08:23,700 --> 00:08:27,730
That corresponds to the
integral from 1 to infinity,

159
00:08:27,730 --> 00:08:30,530
dx / x being infinite.

160
00:08:30,530 --> 00:08:34,810
And the other piece, which --
this one we calculated last

161
00:08:34,810 --> 00:08:37,230
time, this is the one that
we just calculated over here

162
00:08:37,230 --> 00:08:42,719
at Example 2 -- has
the same property.

163
00:08:42,719 --> 00:08:45,720
It's infinite.

164
00:08:45,720 --> 00:08:48,102
And that's the fact that the
integral from 0 to 1 of dx

165
00:08:48,102 --> 00:08:52,100
/ x is infinite.

166
00:08:52,100 --> 00:08:56,570
Right, so both, we
lose on both ends.

167
00:08:56,570 --> 00:09:02,390
On the other hand if I
take something like --

168
00:09:02,390 --> 00:09:05,480
I'm drawing it the same way
but it's really not the same --

169
00:09:05,480 --> 00:09:08,490
y = 1 over the square root of x.

170
00:09:08,490 --> 00:09:11,010
y = 1 / x^(1/2).

171
00:09:11,010 --> 00:09:17,980
And if I cut that in half here
then the x^(1/2) is actually

172
00:09:17,980 --> 00:09:19,890
bigger than this guy.

173
00:09:19,890 --> 00:09:21,740
So this piece is infinite.

174
00:09:21,740 --> 00:09:26,830


175
00:09:26,830 --> 00:09:29,555
And this part over
here actually is going

176
00:09:29,555 --> 00:09:31,310
to give us an honest number.

177
00:09:31,310 --> 00:09:34,810
In fact this one is finite.

178
00:09:34,810 --> 00:09:36,550
And we just checked
what the number is.

179
00:09:36,550 --> 00:09:38,340
It actually happens
to have area 2.

180
00:09:38,340 --> 00:09:46,250


181
00:09:46,250 --> 00:09:49,210
And what's happening here
is if you would superimpose

182
00:09:49,210 --> 00:09:51,710
this graph on the
other graph what you

183
00:09:51,710 --> 00:09:54,580
would see is that they cross.

184
00:09:54,580 --> 00:09:58,970
And this one sits on top.

185
00:09:58,970 --> 00:10:04,660
So if I drew this one in
let's have another color here,

186
00:10:04,660 --> 00:10:05,840
orange let's say.

187
00:10:05,840 --> 00:10:08,630
If this were orange if
I set it on top here

188
00:10:08,630 --> 00:10:11,130
it would go this way.

189
00:10:11,130 --> 00:10:14,790
OK and underneath the
orange is still infinite.

190
00:10:14,790 --> 00:10:16,070
So both of these are infinite.

191
00:10:16,070 --> 00:10:18,028
On here on the other hand
underneath the orange

192
00:10:18,028 --> 00:10:21,700
is infinite but underneath
where the green is is finite.

193
00:10:21,700 --> 00:10:23,770
That's a smaller quantity.

194
00:10:23,770 --> 00:10:25,290
Infinity is a lot bigger than 2.

195
00:10:25,290 --> 00:10:27,826
2 is a lot less than infinity.

196
00:10:27,826 --> 00:10:30,835
All right so that's reflected
in these comparisons here.

197
00:10:30,835 --> 00:10:33,610
Now if you like if I want
to do these in green.

198
00:10:33,610 --> 00:10:39,900
This guy is good and
this guy is good.

199
00:10:39,900 --> 00:10:42,960
Well let me just repeat that
idea over here in this sort

200
00:10:42,960 --> 00:10:47,930
of reversed picture
with y = 1/x^2.

201
00:10:47,930 --> 00:10:53,640
If I chop that in half then
the good part is this end here.

202
00:10:53,640 --> 00:10:54,440
This is finite.

203
00:10:54,440 --> 00:10:56,990


204
00:10:56,990 --> 00:10:59,570
And the bad part is
this part of here

205
00:10:59,570 --> 00:11:01,310
which is way more singular.

206
00:11:01,310 --> 00:11:02,060
And it's infinite.

207
00:11:02,060 --> 00:11:07,070


208
00:11:07,070 --> 00:11:10,080
All right so again what
I've just tried to do

209
00:11:10,080 --> 00:11:16,710
is to give you some
geometric sense and also

210
00:11:16,710 --> 00:11:18,580
some visceral sense.

211
00:11:18,580 --> 00:11:23,040
This guy, its tail as it goes
out to infinity is much lower.

212
00:11:23,040 --> 00:11:25,470
It's much smaller than 1/x.

213
00:11:25,470 --> 00:11:28,080
And these guys trapped an
infinite amount of area.

214
00:11:28,080 --> 00:11:30,110
This one traps only a
finite amount of area.

215
00:11:30,110 --> 00:11:36,676


216
00:11:36,676 --> 00:11:38,860
All right so now I'm
just going to give one

217
00:11:38,860 --> 00:11:43,090
last example which combines
these two types of pictures.

218
00:11:43,090 --> 00:11:45,420
It's really practically
the same as what

219
00:11:45,420 --> 00:11:53,320
I've said before but I-- oh
have to erase this one too.

220
00:11:53,320 --> 00:12:01,340


221
00:12:01,340 --> 00:12:05,770
So here's another example:
if you're in-- So let's

222
00:12:05,770 --> 00:12:08,132
take the following example.

223
00:12:08,132 --> 00:12:09,840
This is somewhat
related to the first one

224
00:12:09,840 --> 00:12:11,600
that I gave last time.

225
00:12:11,600 --> 00:12:16,090
If you take a function
y = 1/(x-3)^2.

226
00:12:16,090 --> 00:12:18,770


227
00:12:18,770 --> 00:12:21,160
And you think
about its integral.

228
00:12:21,160 --> 00:12:24,750
So let's think about the
integral from 0 to infinity,

229
00:12:24,750 --> 00:12:27,090
dx / (x-3)^2.

230
00:12:27,090 --> 00:12:30,420
And suppose you were
faced with this integral.

231
00:12:30,420 --> 00:12:32,630
In order to understand
what it's doing

232
00:12:32,630 --> 00:12:36,310
you have to pay attention to two
places where it can go wrong.

233
00:12:36,310 --> 00:12:39,530
We're going to split
into two pieces.

234
00:12:39,530 --> 00:12:44,349
I'm going say break it up
into this one here up to 5,

235
00:12:44,349 --> 00:12:45,390
for the sake of argument.

236
00:12:45,390 --> 00:12:48,090


237
00:12:48,090 --> 00:12:49,715
And say from 5 to infinity.

238
00:12:49,715 --> 00:12:53,672


239
00:12:53,672 --> 00:12:54,960
All right.

240
00:12:54,960 --> 00:12:56,260
So these are the two chunks.

241
00:12:56,260 --> 00:12:58,810
Now why did I break it
up into those two pieces?

242
00:12:58,810 --> 00:13:00,670
Because what's happening
with this function

243
00:13:00,670 --> 00:13:05,250
is that it's going
up like this at 3.

244
00:13:05,250 --> 00:13:08,210
And so if I look at
the two halves here.

245
00:13:08,210 --> 00:13:09,710
I'm going to draw
them again and I'm

246
00:13:09,710 --> 00:13:12,560
going to illustrate them
with the colors we've chosen,

247
00:13:12,560 --> 00:13:15,350
which are I guess red and green.

248
00:13:15,350 --> 00:13:20,660
What you'll discover
is that this one

249
00:13:20,660 --> 00:13:30,540
here, which corresponds to
this piece here, is infinite.

250
00:13:30,540 --> 00:13:32,320
And it's infinite
because there's

251
00:13:32,320 --> 00:13:34,270
a square in the denominator.

252
00:13:34,270 --> 00:13:39,190
And as x goes to 3 this
is very much like if we

253
00:13:39,190 --> 00:13:40,670
shifted the 3 to 0.

254
00:13:40,670 --> 00:13:42,560
Very much like this 1/x^2 here.

255
00:13:42,560 --> 00:13:44,247
But not in this context.

256
00:13:44,247 --> 00:13:46,330
In the other context where
it's going to infinity.

257
00:13:46,330 --> 00:13:49,460


258
00:13:49,460 --> 00:13:52,360
This is the same
as at the picture

259
00:13:52,360 --> 00:13:57,398
directly above with the
infinite part in red.

260
00:13:57,398 --> 00:13:59,650
All right.

261
00:13:59,650 --> 00:14:04,915
And this part here,
this part is finite.

262
00:14:04,915 --> 00:14:06,240
All right.

263
00:14:06,240 --> 00:14:08,960
So since we have an infinite
part plus a finite part

264
00:14:08,960 --> 00:14:15,070
the conclusion is that this
thing, well this guy converges.

265
00:14:15,070 --> 00:14:18,850
And this one diverges.

266
00:14:18,850 --> 00:14:21,470


267
00:14:21,470 --> 00:14:23,930
But the total
unfortunately diverges.

268
00:14:23,930 --> 00:14:25,760
Right, because it's
got one infinity in it.

269
00:14:25,760 --> 00:14:28,370
So this thing diverges.

270
00:14:28,370 --> 00:14:31,964


271
00:14:31,964 --> 00:14:33,380
And that's what
happened last time

272
00:14:33,380 --> 00:14:34,870
when we got a crazy number.

273
00:14:34,870 --> 00:14:37,480
If you integrated this you
would get some negative number.

274
00:14:37,480 --> 00:14:39,850
If you wrote down the
formulas carelessly.

275
00:14:39,850 --> 00:14:42,250
And the reason is that
the calculation actually

276
00:14:42,250 --> 00:14:44,710
is nonsense.

277
00:14:44,710 --> 00:14:47,890
So you've gotta be
aware, if you encounter

278
00:14:47,890 --> 00:14:51,600
a singularity in the
middle, not to ignore it.

279
00:14:51,600 --> 00:14:52,100
Yeah.

280
00:14:52,100 --> 00:14:52,620
Question.

281
00:14:52,620 --> 00:14:53,786
AUDIENCE: [INAUDIBLE PHRASE]

282
00:14:53,786 --> 00:14:56,430


283
00:14:56,430 --> 00:14:59,710
PROFESSOR: Why do we say that
the whole thing diverges?

284
00:14:59,710 --> 00:15:02,490
The reason why we say that is
the area under the whole curve

285
00:15:02,490 --> 00:15:03,690
is infinite.

286
00:15:03,690 --> 00:15:06,130
It's the sum of this
piece plus this piece.

287
00:15:06,130 --> 00:15:08,218
And so the total is infinite.

288
00:15:08,218 --> 00:15:09,384
AUDIENCE: [INAUDIBLE PHRASE]

289
00:15:09,384 --> 00:15:17,032


290
00:15:17,032 --> 00:15:17,990
PROFESSOR: We're stuck.

291
00:15:17,990 --> 00:15:19,323
This is an ill-defined integral.

292
00:15:19,323 --> 00:15:22,092
It's one where your red flashing
warning sign should be on.

293
00:15:22,092 --> 00:15:23,800
Because you're not
going to get the right

294
00:15:23,800 --> 00:15:24,758
answer by computing it.

295
00:15:24,758 --> 00:15:26,860
You'll never get an answer.

296
00:15:26,860 --> 00:15:29,230
Similarly you'll never
get an answer with this.

297
00:15:29,230 --> 00:15:32,700
And you will get an
answer with that.

298
00:15:32,700 --> 00:15:33,200
OK?

299
00:15:33,200 --> 00:15:37,220


300
00:15:37,220 --> 00:15:38,944
Yeah another question.

301
00:15:38,944 --> 00:15:40,110
AUDIENCE: [INAUDIBLE PHRASE]

302
00:15:40,110 --> 00:15:45,760


303
00:15:45,760 --> 00:15:47,690
PROFESSOR: So the
question is, if you

304
00:15:47,690 --> 00:15:50,010
have a little glance
at an integral,

305
00:15:50,010 --> 00:15:54,750
how are you going to decide
where you should be heading?

306
00:15:54,750 --> 00:15:58,280
So I'm going to
answer that orally.

307
00:15:58,280 --> 00:16:04,010
Although you know, but I'll
say one little hint here.

308
00:16:04,010 --> 00:16:08,070
So you always have to check
x going to infinity and x

309
00:16:08,070 --> 00:16:10,740
going to minus infinity,
if they're in there.

310
00:16:10,740 --> 00:16:15,460
And you also have to check any
singularity, like x going to 3

311
00:16:15,460 --> 00:16:16,419
for sure in this case.

312
00:16:16,419 --> 00:16:18,210
You have to pay attention
to all the places

313
00:16:18,210 --> 00:16:19,376
where the thing is infinite.

314
00:16:19,376 --> 00:16:22,330
And then you want to focus
in on each one separately.

315
00:16:22,330 --> 00:16:26,920
And decide what's going on
it at that particular place.

316
00:16:26,920 --> 00:16:34,530
When it's a negative power-- So
remember dx / x as x goes to 0

317
00:16:34,530 --> 00:16:36,980
is bad.

318
00:16:36,980 --> 00:16:39,510
And dx / x^2 is bad.

319
00:16:39,510 --> 00:16:40,490
dx / x^3 is bad.

320
00:16:40,490 --> 00:16:42,600
All of them are even worse.

321
00:16:42,600 --> 00:16:49,650
So anything of this form
is bad: n = 1, 2, 3.

322
00:16:49,650 --> 00:16:51,980
These are the red box kinds.

323
00:16:51,980 --> 00:16:55,030
All right.

324
00:16:55,030 --> 00:16:56,940
That means that any
of the integrals

325
00:16:56,940 --> 00:16:59,990
that we did in partial
fractions which

326
00:16:59,990 --> 00:17:02,080
had a root, which had
a factor of something

327
00:17:02,080 --> 00:17:03,160
in the denominator.

328
00:17:03,160 --> 00:17:04,790
Those are all
divergent integrals

329
00:17:04,790 --> 00:17:06,670
if you cross the singularly.

330
00:17:06,670 --> 00:17:09,357
Not a single one of them makes
sense across the singularity.

331
00:17:09,357 --> 00:17:09,856
Right?

332
00:17:09,856 --> 00:17:12,692


333
00:17:12,692 --> 00:17:14,150
If you have square
roots and things

334
00:17:14,150 --> 00:17:16,108
like that then you can
repair things like that.

335
00:17:16,108 --> 00:17:18,030
And there's some interesting
examples of that.

336
00:17:18,030 --> 00:17:21,080
Such as with the arcsine
function and so forth.

337
00:17:21,080 --> 00:17:25,940
Where you have an improper
integral which is really OK.

338
00:17:25,940 --> 00:17:26,730
All right.

339
00:17:26,730 --> 00:17:29,880
So that's the best I can do.

340
00:17:29,880 --> 00:17:32,260
It's obviously something
you get experience with.

341
00:17:32,260 --> 00:17:34,110
All right.

342
00:17:34,110 --> 00:17:39,350
Now I'm going to move on
and this is more or less

343
00:17:39,350 --> 00:17:42,150
our last topic.

344
00:17:42,150 --> 00:17:43,900
Yay, but not quite.

345
00:17:43,900 --> 00:17:46,730
Well, so I should say it's
our penultimate topic.

346
00:17:46,730 --> 00:17:49,430
Right because we have
one more lecture.

347
00:17:49,430 --> 00:17:52,010
All right.

348
00:17:52,010 --> 00:17:54,750
So that our next
topic is series.

349
00:17:54,750 --> 00:17:58,810
Now we'll do it in a sort
of a concrete way today.

350
00:17:58,810 --> 00:18:02,775
And then we'll do what are
known as power series tomorrow.

351
00:18:02,775 --> 00:18:05,400


352
00:18:05,400 --> 00:18:06,760
So let me tell you about series.

353
00:18:06,760 --> 00:18:20,490


354
00:18:20,490 --> 00:18:22,604
Remember we're
talking about infinity

355
00:18:22,604 --> 00:18:23,687
and dealing with infinity.

356
00:18:23,687 --> 00:18:26,772


357
00:18:26,772 --> 00:18:28,730
So we're not just talking
about any old series.

358
00:18:28,730 --> 00:18:30,230
We're talking about
infinite series.

359
00:18:30,230 --> 00:18:32,650


360
00:18:32,650 --> 00:18:37,400
There is one infinite
series which is probably,

361
00:18:37,400 --> 00:18:39,000
which is without
question the most

362
00:18:39,000 --> 00:18:41,980
important and useful series.

363
00:18:41,980 --> 00:18:44,690
And that's the
geometric series but I'm

364
00:18:44,690 --> 00:18:48,405
going to introduce it concretely
first in a particular case.

365
00:18:48,405 --> 00:18:51,990


366
00:18:51,990 --> 00:18:54,260
If I draw a picture of this sum.

367
00:18:54,260 --> 00:18:56,480
Which in principle
goes on forever.

368
00:18:56,480 --> 00:18:59,670
You can see that it goes
someplace fairly easily

369
00:18:59,670 --> 00:19:02,730
by marking out what's
happening on the number line.

370
00:19:02,730 --> 00:19:06,630
The first step takes
us to 1 from 0.

371
00:19:06,630 --> 00:19:11,790
And then if I add this
half, I get to 3/2.

372
00:19:11,790 --> 00:19:16,240
Right, so the first step was
1 and the second step was 1/2.

373
00:19:16,240 --> 00:19:20,940
Now if I add this quarter in,
which is the next piece then

374
00:19:20,940 --> 00:19:22,400
I get some place here.

375
00:19:22,400 --> 00:19:28,440
But what I want to
observe is that I got,

376
00:19:28,440 --> 00:19:31,130
I can look at it from
the other point of view.

377
00:19:31,130 --> 00:19:36,240
I got, when I move this quarter
I got half way to 2 here.

378
00:19:36,240 --> 00:19:39,420
I'm putting 2 in green
because I want you to think

379
00:19:39,420 --> 00:19:42,950
of it as being the good kind.

380
00:19:42,950 --> 00:19:43,970
Right.

381
00:19:43,970 --> 00:19:45,420
The kind that has a number.

382
00:19:45,420 --> 00:19:47,280
And not one of the red kinds.

383
00:19:47,280 --> 00:19:49,920
We're getting there
and we're almost there.

384
00:19:49,920 --> 00:19:52,620
So the next stage we
get half way again.

385
00:19:52,620 --> 00:19:54,605
That's the eighth and so forth.

386
00:19:54,605 --> 00:19:56,490
And eventually we get to 2.

387
00:19:56,490 --> 00:20:00,476
So this sum we write equals two.

388
00:20:00,476 --> 00:20:02,590
All right that's
kind of a paradox

389
00:20:02,590 --> 00:20:04,130
because we never get to 2.

390
00:20:04,130 --> 00:20:08,080
This is the paradox
that Zeno fussed with.

391
00:20:08,080 --> 00:20:12,807
And his conclusion, you know,
with the rabbit and the hare.

392
00:20:12,807 --> 00:20:14,140
No, the rabbit and the tortoise.

393
00:20:14,140 --> 00:20:18,890
Sorry hare chasing-- anyway,
the rabbit chasing the tortoise.

394
00:20:18,890 --> 00:20:21,050
His conclusion--
you know, I don't

395
00:20:21,050 --> 00:20:23,770
know if you're aware of this,
but he understood this paradox.

396
00:20:23,770 --> 00:20:25,811
And he said you know it
doesn't look like it ever

397
00:20:25,811 --> 00:20:29,180
gets there because they're
infinitely many times

398
00:20:29,180 --> 00:20:32,330
between the time-- you know that
the tortoise is always behind,

399
00:20:32,330 --> 00:20:34,580
always behind, always
behind, always behind.

400
00:20:34,580 --> 00:20:36,970
So therefore it's
impossible that the tortoise

401
00:20:36,970 --> 00:20:38,710
catches up right.

402
00:20:38,710 --> 00:20:41,600
So do you know what
his conclusion was?

403
00:20:41,600 --> 00:20:45,280
Time does not exist.

404
00:20:45,280 --> 00:20:48,007
That was actually
literally his conclusion.

405
00:20:48,007 --> 00:20:49,840
Because he didn't
understand the possibility

406
00:20:49,840 --> 00:20:50,799
of a continuum of time.

407
00:20:50,799 --> 00:20:53,090
Because there were infinitely
many things that happened

408
00:20:53,090 --> 00:20:56,610
before the tortoise caught up.

409
00:20:56,610 --> 00:20:57,790
So that was the reasoning.

410
00:20:57,790 --> 00:21:00,430
I mean it's a long time ago
but you know people didn't-- he

411
00:21:00,430 --> 00:21:02,430
didn't believe in continuum.

412
00:21:02,430 --> 00:21:03,480
All right.

413
00:21:03,480 --> 00:21:06,940
So anyway that's a small point.

414
00:21:06,940 --> 00:21:19,410
Now the general case here
of a geometric series

415
00:21:19,410 --> 00:21:22,820
is where I put in a number
a instead of 1/2 here.

416
00:21:22,820 --> 00:21:23,870
So what we had before.

417
00:21:23,870 --> 00:21:26,890
So that's 1 + a + a^2...

418
00:21:26,890 --> 00:21:31,900
Isn't quite the most general
but anyway I'll write this down.

419
00:21:31,900 --> 00:21:34,630
And you're certainly going
to want to remember that

420
00:21:34,630 --> 00:21:39,510
the formula for this in
the limit is 1/(1-a).

421
00:21:39,510 --> 00:21:44,000
And I remind you that this only
works when the absolute value

422
00:21:44,000 --> 00:21:45,810
is strictly less than 1.

423
00:21:45,810 --> 00:21:47,970
In other words when -1
is strictly less than a

424
00:21:47,970 --> 00:21:51,130
is less than 1.

425
00:21:51,130 --> 00:21:53,090
And that's really the
issue that we're going

426
00:21:53,090 --> 00:21:54,310
to want to worry about now.

427
00:21:54,310 --> 00:21:58,270
What we're worrying about is
this notion of convergence.

428
00:21:58,270 --> 00:22:05,300
And what goes wrong when
there isn't convergence,

429
00:22:05,300 --> 00:22:07,360
when there's a divergence.

430
00:22:07,360 --> 00:22:13,220
So let me illustrate the
divergences before going on.

431
00:22:13,220 --> 00:22:15,900
And this is what we
have to avoid if we're

432
00:22:15,900 --> 00:22:18,990
going to understand series.

433
00:22:18,990 --> 00:22:21,890
So here's an example when a = 1.

434
00:22:21,890 --> 00:22:26,620
You get 1 + 1 +
1 plus et cetera.

435
00:22:26,620 --> 00:22:29,990
And that's equal to 1/(1-1).

436
00:22:29,990 --> 00:22:32,370
Which is 1 over 0.

437
00:22:32,370 --> 00:22:33,760
So this is not bad.

438
00:22:33,760 --> 00:22:34,861
It's almost right.

439
00:22:34,861 --> 00:22:35,360
Right?

440
00:22:35,360 --> 00:22:37,750
It's sort of infinity
equals infinity.

441
00:22:37,750 --> 00:22:39,850
At the edge here we
managed to get something

442
00:22:39,850 --> 00:22:42,340
which is sort of almost right.

443
00:22:42,340 --> 00:22:46,100
But you know, it's, we don't
consider this to be logically

444
00:22:46,100 --> 00:22:47,540
to make complete sense.

445
00:22:47,540 --> 00:22:51,200
So it's a little dangerous.

446
00:22:51,200 --> 00:22:52,960
And so we just say
that it diverges.

447
00:22:52,960 --> 00:22:54,100
And we get rid of this.

448
00:22:54,100 --> 00:22:55,990
So we're still
putting it in red.

449
00:22:55,990 --> 00:22:58,540
All right.

450
00:22:58,540 --> 00:22:59,975
The bad guy here.

451
00:22:59,975 --> 00:23:00,850
So this one diverges.

452
00:23:00,850 --> 00:23:04,670


453
00:23:04,670 --> 00:23:12,650
Similarly if I take a equals
-1, I get 1 - 1 + 1 - 1 + 1...

454
00:23:12,650 --> 00:23:15,450
Because the odd and the
even powers in that formula

455
00:23:15,450 --> 00:23:17,270
alternate sign.

456
00:23:17,270 --> 00:23:19,750
And this bounces back and forth.

457
00:23:19,750 --> 00:23:21,640
It never settles down.

458
00:23:21,640 --> 00:23:23,346
It starts at 1.

459
00:23:23,346 --> 00:23:25,720
And then it gets down to 0
and then it goes back up to 1,

460
00:23:25,720 --> 00:23:28,400
down to 0, back up to 1.

461
00:23:28,400 --> 00:23:29,517
It doesn't settle down.

462
00:23:29,517 --> 00:23:30,600
It bounces back and forth.

463
00:23:30,600 --> 00:23:31,580
It oscillates.

464
00:23:31,580 --> 00:23:34,740
On the other hand if you
compare the right hand side.

465
00:23:34,740 --> 00:23:35,980
What's the right hand side?

466
00:23:35,980 --> 00:23:36,970
It's 1 / (1-(-1)).

467
00:23:36,970 --> 00:23:39,730


468
00:23:39,730 --> 00:23:41,515
Which is 1/2.

469
00:23:41,515 --> 00:23:42,460
All right.

470
00:23:42,460 --> 00:23:45,169
So if you just paid attention
to the formula, which

471
00:23:45,169 --> 00:23:47,710
is what we were doing when we
integrated without thinking too

472
00:23:47,710 --> 00:23:50,070
hard about this,
you get a number

473
00:23:50,070 --> 00:23:51,320
here but in fact that's wrong.

474
00:23:51,320 --> 00:23:53,153
Actually it's kind of
an interesting number.

475
00:23:53,153 --> 00:23:56,380
It's halfway between the
two, between 0 and 1.

476
00:23:56,380 --> 00:23:58,505
So again there's some
sort of vague sense

477
00:23:58,505 --> 00:24:01,966
in which this is trying
to be this answer.

478
00:24:01,966 --> 00:24:04,580
All right.

479
00:24:04,580 --> 00:24:08,744
It's not so bad but we're still
going to put this in a red box.

480
00:24:08,744 --> 00:24:10,030
All right.

481
00:24:10,030 --> 00:24:12,710
because this is what
we called divergence.

482
00:24:12,710 --> 00:24:16,130
So both of these
cases are divergent.

483
00:24:16,130 --> 00:24:20,490
It only really works when
alpha-- when a is less than 1.

484
00:24:20,490 --> 00:24:23,480
I'm going to add
one more case just

485
00:24:23,480 --> 00:24:30,040
to see that mathematicians are
slightly curious about what

486
00:24:30,040 --> 00:24:32,020
goes on in other cases.

487
00:24:32,020 --> 00:24:37,570
So this is 1 + 2 +
2^2 + 2^3 plus etc..

488
00:24:37,570 --> 00:24:41,980
And that should be equal to --
according to this formula --

489
00:24:41,980 --> 00:24:44,750
1/(1-2).

490
00:24:44,750 --> 00:24:48,242
Which is -1.

491
00:24:48,242 --> 00:24:49,860
All right.

492
00:24:49,860 --> 00:24:53,460
Now this one is
clearly wrong, right?

493
00:24:53,460 --> 00:24:55,530
This one is totally wrong.

494
00:24:55,530 --> 00:24:58,170


495
00:24:58,170 --> 00:24:59,465
It certainly diverges.

496
00:24:59,465 --> 00:25:02,370
The left hand side is
obviously infinite.

497
00:25:02,370 --> 00:25:04,070
The right hand side is way off.

498
00:25:04,070 --> 00:25:05,960
It's -1.

499
00:25:05,960 --> 00:25:10,350
On the other hand it
turns out actually

500
00:25:10,350 --> 00:25:13,360
that mathematicians have ways
of making sense out of these.

501
00:25:13,360 --> 00:25:15,590
In number theory
there's a strange system

502
00:25:15,590 --> 00:25:18,160
where this is actually true.

503
00:25:18,160 --> 00:25:21,530
And what happens
in that system is

504
00:25:21,530 --> 00:25:24,030
that what you have to
throw out is the idea

505
00:25:24,030 --> 00:25:27,050
that 0 is less than 1.

506
00:25:27,050 --> 00:25:29,980
There is no such thing
as negative numbers.

507
00:25:29,980 --> 00:25:32,090
So this number exists.

508
00:25:32,090 --> 00:25:35,700
And it's the additive
inverse of 1.

509
00:25:35,700 --> 00:25:39,780
It has this arithmetic
property but the statement

510
00:25:39,780 --> 00:25:43,330
that this is, that 1 is bigger
than 0 does not make sense.

511
00:25:43,330 --> 00:25:45,440
So you have your choice,
either this diverges

512
00:25:45,440 --> 00:25:48,630
or you have to throw
out something like this.

513
00:25:48,630 --> 00:25:51,510
So that's a very curious
thing in higher mathematics.

514
00:25:51,510 --> 00:25:56,410
Which if you get to number
theory there's fun stuff there.

515
00:25:56,410 --> 00:25:58,920
All right.

516
00:25:58,920 --> 00:26:02,740
OK but for our purposes
these things are all out.

517
00:26:02,740 --> 00:26:03,300
All right.

518
00:26:03,300 --> 00:26:04,010
They're gone.

519
00:26:04,010 --> 00:26:05,160
We're not considering them.

520
00:26:05,160 --> 00:26:09,550
Only a between -1 and 1.

521
00:26:09,550 --> 00:26:10,070
All right.

522
00:26:10,070 --> 00:26:13,910


523
00:26:13,910 --> 00:26:18,190
Now I want to do
something systematic.

524
00:26:18,190 --> 00:26:21,530
And it's more or less on
the lines of the powers

525
00:26:21,530 --> 00:26:23,090
that I'm erasing right now.

526
00:26:23,090 --> 00:26:26,630


527
00:26:26,630 --> 00:26:28,590
I want to tell you
about series which are

528
00:26:28,590 --> 00:26:30,420
kind of borderline convergent.

529
00:26:30,420 --> 00:26:33,720
And then next time when we
talk about powers series we'll

530
00:26:33,720 --> 00:26:35,810
come back to this very
important series which

531
00:26:35,810 --> 00:26:37,080
is the most important one.

532
00:26:37,080 --> 00:26:40,680


533
00:26:40,680 --> 00:26:47,400
So now let's talk about some
series-- er, general notations.

534
00:26:47,400 --> 00:26:49,970
And this will help
you with the last bit.

535
00:26:49,970 --> 00:26:53,810


536
00:26:53,810 --> 00:26:56,760
This is going to be pretty
much the same as what

537
00:26:56,760 --> 00:27:00,890
we did for improper integrals.

538
00:27:00,890 --> 00:27:04,320
Namely, first of all I'm
going to have S_N which

539
00:27:04,320 --> 00:27:09,760
is the sum of a_n, n
equals 0 to capital N.

540
00:27:09,760 --> 00:27:12,220
And this is what we're
calling a partial sum.

541
00:27:12,220 --> 00:27:18,050


542
00:27:18,050 --> 00:27:23,980
And then the full limit, which
is capital S, if you like.

543
00:27:23,980 --> 00:27:29,980
a_n, n equals 0 to infinity,
is just the limit as N goes

544
00:27:29,980 --> 00:27:32,010
to infinity of the S_N's.

545
00:27:32,010 --> 00:27:36,240


546
00:27:36,240 --> 00:27:39,770
And then we have the same kind
of notation that we had before.

547
00:27:39,770 --> 00:27:42,860
Which is there are
these two choices which

548
00:27:42,860 --> 00:27:45,890
is that if the limit exists.

549
00:27:45,890 --> 00:27:50,450


550
00:27:50,450 --> 00:27:51,830
That's the green choice.

551
00:27:51,830 --> 00:27:54,460
And we say it converges.

552
00:27:54,460 --> 00:28:00,830
So we say the series converges.

553
00:28:00,830 --> 00:28:06,616
And then the other case which
is the limit does not exist.

554
00:28:06,616 --> 00:28:10,782


555
00:28:10,782 --> 00:28:12,240
And we can say the
series diverges.

556
00:28:12,240 --> 00:28:20,560


557
00:28:20,560 --> 00:28:21,343
Question.

558
00:28:21,343 --> 00:28:22,509
AUDIENCE: [INAUDIBLE PHRASE]

559
00:28:22,509 --> 00:28:26,480


560
00:28:26,480 --> 00:28:29,290
PROFESSOR: The question
was how did I get to this?

561
00:28:29,290 --> 00:28:31,840
And I will do that next
time but in fact of course

562
00:28:31,840 --> 00:28:33,240
you've seen it in high school.

563
00:28:33,240 --> 00:28:35,930
Right this is-- Yeah.

564
00:28:35,930 --> 00:28:36,860
Yeah.

565
00:28:36,860 --> 00:28:40,000
We'll do that next time.

566
00:28:40,000 --> 00:28:42,222
The question was how
did we arrive-- sorry I

567
00:28:42,222 --> 00:28:43,430
didn't tell you the question.

568
00:28:43,430 --> 00:28:44,360
The question was
how do we arrive

569
00:28:44,360 --> 00:28:46,590
at this formula on the
right hand side here.

570
00:28:46,590 --> 00:28:48,240
But we'll talk about
that next time.

571
00:28:48,240 --> 00:28:53,060


572
00:28:53,060 --> 00:28:54,020
All right.

573
00:28:54,020 --> 00:28:59,930
So here's the basic
definition and what

574
00:28:59,930 --> 00:29:02,430
we're going to
recognize about series.

575
00:29:02,430 --> 00:29:06,835
And I'm going to give
you a few examples

576
00:29:06,835 --> 00:29:08,460
and then we'll do
something systematic.

577
00:29:08,460 --> 00:29:12,070


578
00:29:12,070 --> 00:29:14,495
So the first example--
well the first example

579
00:29:14,495 --> 00:29:16,290
is the geometric series.

580
00:29:16,290 --> 00:29:19,430
But the first example that
I'm going to discuss now

581
00:29:19,430 --> 00:29:23,352
and in a little bit of
detail is this sum 1/n^2,

582
00:29:23,352 --> 00:29:24,310
n equals 1 to infinity.

583
00:29:24,310 --> 00:29:28,580


584
00:29:28,580 --> 00:29:34,050
It turns out that this
series is very analogous --

585
00:29:34,050 --> 00:29:36,690
and we'll develop this
analogy carefully --

586
00:29:36,690 --> 00:29:41,065
the integral from
1 to x, dx / x^2.

587
00:29:41,065 --> 00:29:46,220
And we're going to develop
this analogy in detail later

588
00:29:46,220 --> 00:29:47,920
in this lecture.

589
00:29:47,920 --> 00:29:49,770
And this one is
one of the ones--

590
00:29:49,770 --> 00:29:51,890
so now you have to go back
and actually remember,

591
00:29:51,890 --> 00:29:54,335
this is one of the ones you
really want to memorize.

592
00:29:54,335 --> 00:29:56,460
And you should especially
pay attention to the ones

593
00:29:56,460 --> 00:29:58,507
with an infinity in them.

594
00:29:58,507 --> 00:29:59,465
This one is convergent.

595
00:29:59,465 --> 00:30:03,270


596
00:30:03,270 --> 00:30:04,520
And this series is convergent.

597
00:30:04,520 --> 00:30:11,070
Now it turns out that
evaluating this is very easy.

598
00:30:11,070 --> 00:30:12,820
This is 1.

599
00:30:12,820 --> 00:30:15,310
It's easy to calculate.

600
00:30:15,310 --> 00:30:19,500
Evaluating this is very tricky.

601
00:30:19,500 --> 00:30:21,610
And Euler did it.

602
00:30:21,610 --> 00:30:26,440
And the answer is pi^2 / 6.

603
00:30:26,440 --> 00:30:29,050
That's an amazing calculation.

604
00:30:29,050 --> 00:30:33,570
And it was done very early in
the history of mathematics.

605
00:30:33,570 --> 00:30:38,110
If you look at another example--
so maybe example two here,

606
00:30:38,110 --> 00:30:45,790
if you look at 1/n^3, n equals--
well you can't start here at 0

607
00:30:45,790 --> 00:30:46,580
by the way.

608
00:30:46,580 --> 00:30:48,930
I get to start wherever
I want in these series.

609
00:30:48,930 --> 00:30:49,920
Here I start with 0.

610
00:30:49,920 --> 00:30:51,220
Here I started with 1.

611
00:30:51,220 --> 00:30:52,880
And notice the reason
why I started--

612
00:30:52,880 --> 00:30:56,650
it was a bad idea to start
with 0 was that 1 over 0

613
00:30:56,650 --> 00:30:57,860
is undefined.

614
00:30:57,860 --> 00:30:58,360
Right?

615
00:30:58,360 --> 00:31:00,443
So I'm just starting where
it's convenient for me.

616
00:31:00,443 --> 00:31:03,560
And since I'm interested
mostly in the tail behavior

617
00:31:03,560 --> 00:31:06,150
it doesn't matter to me
so much where I start.

618
00:31:06,150 --> 00:31:07,800
Although if I want
an exact answer

619
00:31:07,800 --> 00:31:10,165
I need to start
exactly at n = 1.

620
00:31:10,165 --> 00:31:10,890
All right.

621
00:31:10,890 --> 00:31:17,932
This one is similar
to this integral here.

622
00:31:17,932 --> 00:31:18,700
All right.

623
00:31:18,700 --> 00:31:20,440
Which is convergent again.

624
00:31:20,440 --> 00:31:22,390
So there's a number
that you get.

625
00:31:22,390 --> 00:31:27,180
And let's see what is it
something like 2/3 or something

626
00:31:27,180 --> 00:31:30,190
like that, all right,
for this for this number.

627
00:31:30,190 --> 00:31:32,440
Or 1/3.

628
00:31:32,440 --> 00:31:33,110
What is it?

629
00:31:33,110 --> 00:31:33,690
No 1/2.

630
00:31:33,690 --> 00:31:35,000
I guess it's 1/2.

631
00:31:35,000 --> 00:31:37,110
This one is 1/2.

632
00:31:37,110 --> 00:31:38,780
You check that,
I'm not positive,

633
00:31:38,780 --> 00:31:40,600
but anyway just doing
it in my head quickly

634
00:31:40,600 --> 00:31:42,150
it seems to be 1/2.

635
00:31:42,150 --> 00:31:44,100
Anyway it's an easy
number to calculate.

636
00:31:44,100 --> 00:31:49,710
This one over here stumped
mathematicians basically

637
00:31:49,710 --> 00:31:51,870
for all time.

638
00:31:51,870 --> 00:31:56,250
It doesn't have any kind of
elementary form like this.

639
00:31:56,250 --> 00:31:59,842
And it was only very recently
proved to be rational.

640
00:31:59,842 --> 00:32:01,550
People couldn't even
couldn't even decide

641
00:32:01,550 --> 00:32:05,260
whether this was a
rational number or not.

642
00:32:05,260 --> 00:32:07,830
But anyway that's been resolved;
it is an irrational number

643
00:32:07,830 --> 00:32:09,540
which is what people suspected.

644
00:32:09,540 --> 00:32:10,650
Yeah question.

645
00:32:10,650 --> 00:32:11,816
AUDIENCE: [INAUDIBLE PHRASE]

646
00:32:11,816 --> 00:32:14,360


647
00:32:14,360 --> 00:32:16,220
PROFESSOR: Yeah sorry.

648
00:32:16,220 --> 00:32:16,720
OK.

649
00:32:16,720 --> 00:32:19,820


650
00:32:19,820 --> 00:32:25,070
I violated a rule
of mathematics--

651
00:32:25,070 --> 00:32:26,720
you said why is this similar?

652
00:32:26,720 --> 00:32:29,140
I thought that similar
was something else.

653
00:32:29,140 --> 00:32:30,380
And you're absolutely right.

654
00:32:30,380 --> 00:32:33,520
And I violated a
rule of mathematics.

655
00:32:33,520 --> 00:32:37,825
Which is that I used this
symbol for two different things.

656
00:32:37,825 --> 00:32:41,196


657
00:32:41,196 --> 00:32:42,820
I should have written
this symbol here.

658
00:32:42,820 --> 00:32:43,530
All right.

659
00:32:43,530 --> 00:32:45,260
I'll create a new symbol here.

660
00:32:45,260 --> 00:32:48,920
The question of whether this
converges or this converges.

661
00:32:48,920 --> 00:32:51,920
These are the the
same type of question.

662
00:32:51,920 --> 00:32:54,090
And we'll see why they're
the same question it

663
00:32:54,090 --> 00:32:55,000
in a few minutes.

664
00:32:55,000 --> 00:32:58,190
But in fact the wiggle
I used, "similar",

665
00:32:58,190 --> 00:33:02,360
I used for the connection
between functions.

666
00:33:02,360 --> 00:33:09,070
The things that are really
similar are that 1/n resembles

667
00:33:09,070 --> 00:33:10,230
1/x^2.

668
00:33:10,230 --> 00:33:12,066
So I apologize I didn't--

669
00:33:12,066 --> 00:33:13,232
AUDIENCE: [INAUDIBLE PHRASE]

670
00:33:13,232 --> 00:33:15,890


671
00:33:15,890 --> 00:33:17,890
PROFESSOR: Oh you
thought that this

672
00:33:17,890 --> 00:33:19,210
was the definition of that.

673
00:33:19,210 --> 00:33:21,460
That's actually the reason
why these things correspond

674
00:33:21,460 --> 00:33:21,990
so closely.

675
00:33:21,990 --> 00:33:25,560
That is that the Riemann
sum is close to this.

676
00:33:25,560 --> 00:33:27,520
But that doesn't
mean they're equal.

677
00:33:27,520 --> 00:33:31,245
The Riemann sum only works
when the delta x goes to 0.

678
00:33:31,245 --> 00:33:33,870
The way that we're going to get
a connection between these two,

679
00:33:33,870 --> 00:33:38,710
as we will just a second, is
with a Riemann sum with-- What

680
00:33:38,710 --> 00:33:47,180
we're going to use is a
Riemann sum with delta x = 1.

681
00:33:47,180 --> 00:33:49,617
All right and then that will
be the connection between.

682
00:33:49,617 --> 00:33:53,240
All right that's
absolutely right.

683
00:33:53,240 --> 00:33:53,740
All right.

684
00:33:53,740 --> 00:33:57,020


685
00:33:57,020 --> 00:34:00,280
So in order to illustrate
exactly this idea

686
00:34:00,280 --> 00:34:02,110
that you've just come
up with, and in fact

687
00:34:02,110 --> 00:34:04,490
that we're going to use,
we'll do the same thing

688
00:34:04,490 --> 00:34:07,740
but we're going to do it
on the example sum 1/n.

689
00:34:07,740 --> 00:34:13,380


690
00:34:13,380 --> 00:34:20,170
So here's Example 3 and
it's going to be sum 1/n,

691
00:34:20,170 --> 00:34:21,369
n equals 1 to infinity.

692
00:34:21,369 --> 00:34:24,090


693
00:34:24,090 --> 00:34:27,040
And what we're now
going to see is

694
00:34:27,040 --> 00:34:29,340
that it corresponds
to this integral here.

695
00:34:29,340 --> 00:34:32,620


696
00:34:32,620 --> 00:34:34,850
And we're going
to show therefore

697
00:34:34,850 --> 00:34:37,870
that this thing diverges.

698
00:34:37,870 --> 00:34:40,300
But we're going to do
this more carefully.

699
00:34:40,300 --> 00:34:42,750
We're going to do
this in some detail

700
00:34:42,750 --> 00:34:45,880
so that you see what it is,
that the correspondence is

701
00:34:45,880 --> 00:34:47,390
between these quantities.

702
00:34:47,390 --> 00:34:49,890
And the same sort
of reasoning applies

703
00:34:49,890 --> 00:34:51,275
to these other examples.

704
00:34:51,275 --> 00:34:55,820


705
00:34:55,820 --> 00:34:59,180
So here we go.

706
00:34:59,180 --> 00:35:04,290
I'm going to take
the integral and draw

707
00:35:04,290 --> 00:35:07,220
the picture of the Riemann sum.

708
00:35:07,220 --> 00:35:13,840
So here's the level 1 and
here's the function y = 1/x.

709
00:35:13,840 --> 00:35:15,730
And I'm going to
take the Riemann sum.

710
00:35:15,730 --> 00:35:21,980


711
00:35:21,980 --> 00:35:25,200
With delta x = 1.

712
00:35:25,200 --> 00:35:28,820
And that's going to be closely
connected to the series

713
00:35:28,820 --> 00:35:29,570
that I have.

714
00:35:29,570 --> 00:35:32,160


715
00:35:32,160 --> 00:35:35,670
But now I have to decide
whether I want a lower Riemann

716
00:35:35,670 --> 00:35:37,797
sum or an upper Riemann sum.

717
00:35:37,797 --> 00:35:39,630
And actually I'm going
to check both of them

718
00:35:39,630 --> 00:35:41,213
because both of them
are illuminating.

719
00:35:41,213 --> 00:35:44,620


720
00:35:44,620 --> 00:35:47,000
First we'll do the
upper Riemann's sum.

721
00:35:47,000 --> 00:35:48,700
Now that's this staircase here.

722
00:35:48,700 --> 00:35:51,780


723
00:35:51,780 --> 00:35:54,600
So we'll call this the
upper Riemann's sum.

724
00:35:54,600 --> 00:35:58,130


725
00:35:58,130 --> 00:35:59,710
And let's check
what its levels are.

726
00:35:59,710 --> 00:36:01,560
This is not to scale.

727
00:36:01,560 --> 00:36:03,250
This level should be 1/2.

728
00:36:03,250 --> 00:36:05,245
So if this is 1 and
this is 2 and that level

729
00:36:05,245 --> 00:36:10,340
was supposed to be 1/2 and
this next level should be 1/3.

730
00:36:10,340 --> 00:36:12,830
That's how the Riemann
sums are working out.

731
00:36:12,830 --> 00:36:17,040


732
00:36:17,040 --> 00:36:21,770
And now I have the
following phenomenon.

733
00:36:21,770 --> 00:36:24,640
Let's cut it off
at the nth stage.

734
00:36:24,640 --> 00:36:27,095
So that means that I'm
going, the integral

735
00:36:27,095 --> 00:36:30,760
is from 1 to n, dx / x.

736
00:36:30,760 --> 00:36:33,690
And the Riemann sum is
something that's bigger than it.

737
00:36:33,690 --> 00:36:40,300
Because the areas are enclosing
the area of the curved region.

738
00:36:40,300 --> 00:36:43,510
And that's going to be the
area of the first box which

739
00:36:43,510 --> 00:36:50,330
is 1, plus the area of the
second box which is 1/2,

740
00:36:50,330 --> 00:36:54,150
plus the area of the
third box which is 1/3.

741
00:36:54,150 --> 00:37:00,840
All the way up the last one,
but the last one starts at N-1.

742
00:37:00,840 --> 00:37:03,450
So it has 1/(N-1).

743
00:37:03,450 --> 00:37:05,780
There are not N boxes here.

744
00:37:05,780 --> 00:37:07,960
There are only N-1 boxes.

745
00:37:07,960 --> 00:37:11,830
Because the distance
between 1 and N is N-1.

746
00:37:11,830 --> 00:37:13,555
Right so this is N-1 terms.

747
00:37:13,555 --> 00:37:17,330


748
00:37:17,330 --> 00:37:25,140
However, if I use the
notation for partial sum.

749
00:37:25,140 --> 00:37:33,367
Which is 1 + 1/2 plus all the
way up to 1/(n-1) 1 + 1/n.

750
00:37:33,367 --> 00:37:35,200
In other words I go out
to the Nth one which

751
00:37:35,200 --> 00:37:37,370
is what I would ordinarily do.

752
00:37:37,370 --> 00:37:42,900
Then this sum that I have here
certainly is less than S_N.

753
00:37:42,900 --> 00:37:47,850
Because there's one
more term there.

754
00:37:47,850 --> 00:37:50,430
And so here I have
an integral which

755
00:37:50,430 --> 00:37:53,570
is underneath this sum S_N.

756
00:37:53,570 --> 00:38:01,700


757
00:38:01,700 --> 00:38:19,031
Now this is going to allow
us to prove conclusively

758
00:38:19,031 --> 00:38:21,156
that the-- So I'm just
going to rewrite this, prove

759
00:38:21,156 --> 00:38:22,614
conclusively that
the sum diverges.

760
00:38:22,614 --> 00:38:23,340
Why is that?

761
00:38:23,340 --> 00:38:26,000
Because this term
here we can calculate.

762
00:38:26,000 --> 00:38:29,470
This is log x
evaluated at 1 and n.

763
00:38:29,470 --> 00:38:34,850
Which is the same
thing as log N minus 0.

764
00:38:34,850 --> 00:38:39,270
All right, the quantity
log N - log 1 which is 0.

765
00:38:39,270 --> 00:38:46,892
And so what we have here is
that log N is less than S_N.

766
00:38:46,892 --> 00:38:51,455
All right and clearly this
goes to infinity right.

767
00:38:51,455 --> 00:38:57,830
As N goes to infinity this
thing goes to infinity.

768
00:38:57,830 --> 00:38:58,435
So we're done.

769
00:38:58,435 --> 00:39:00,330
All right we've
shown divergence.

770
00:39:00,330 --> 00:39:08,730


771
00:39:08,730 --> 00:39:15,040
Now the way I'm going to
use the lower Riemann's sum

772
00:39:15,040 --> 00:39:20,010
is to recognize
that we've captured

773
00:39:20,010 --> 00:39:21,960
the rate appropriately.

774
00:39:21,960 --> 00:39:24,400
That is not only do I have
a lower bound like this

775
00:39:24,400 --> 00:39:27,860
but I have an upper bound
which is very similar.

776
00:39:27,860 --> 00:39:30,440
So if I use the upper
Riemann-- oh sorry,

777
00:39:30,440 --> 00:39:39,540
lower Riemann sum
again with delta x = 1.

778
00:39:39,540 --> 00:39:43,760


779
00:39:43,760 --> 00:39:53,946
Then I have that the
integral from 1 to n of dx

780
00:39:53,946 --> 00:39:57,970
/ x is bigger than-- Well
what are the terms going

781
00:39:57,970 --> 00:40:00,640
to be if fit them underneath?

782
00:40:00,640 --> 00:40:03,210
If I fit them underneath
I'm missing the first term.

783
00:40:03,210 --> 00:40:05,600
That is the box is
going to be half height.

784
00:40:05,600 --> 00:40:08,300
It's going to be
this lower piece.

785
00:40:08,300 --> 00:40:10,480
So I'm missing this first term.

786
00:40:10,480 --> 00:40:16,765
So it'll be a 1/2 + 1/3 plus...

787
00:40:16,765 --> 00:40:18,150
All right, it will
keep on going.

788
00:40:18,150 --> 00:40:22,150
But now the last one
instead of being 1/(N-1),

789
00:40:22,150 --> 00:40:25,850
it's going to be 1 over N. This
is again a total of the N-1

790
00:40:25,850 --> 00:40:27,050
terms.

791
00:40:27,050 --> 00:40:28,430
This is the lower Riemann sum.

792
00:40:28,430 --> 00:40:31,190


793
00:40:31,190 --> 00:40:38,808
And now we can recognize that
this is exactly equal to-- well

794
00:40:38,808 --> 00:40:40,558
so I'll put it over
here-- this is exactly

795
00:40:40,558 --> 00:40:43,520
equal to S_N minus 1,
minus the first term.

796
00:40:43,520 --> 00:40:47,110
So we missed the first term but
we got all the rest of them.

797
00:40:47,110 --> 00:40:49,360
So if I put this
to the other side

798
00:40:49,360 --> 00:40:53,746
remember this is
log N. All right.

799
00:40:53,746 --> 00:40:55,120
If I put this to
the other side I

800
00:40:55,120 --> 00:40:57,130
have the other
side of this bound.

801
00:40:57,130 --> 00:41:07,010
I have that S S_N is less than,
if I reverse it, log N + 1.

802
00:41:07,010 --> 00:41:09,040
And so I've trapped
it on the other side.

803
00:41:09,040 --> 00:41:10,950
And here I have the lower bound.

804
00:41:10,950 --> 00:41:13,170
So I'm going to
combine those together.

805
00:41:13,170 --> 00:41:18,030
So all told I have this
correspondence here.

806
00:41:18,030 --> 00:41:23,150
It is the size of log N is
trapped between the-- sorry,

807
00:41:23,150 --> 00:41:26,150
the size of S_N, which is
relatively hard to calculate

808
00:41:26,150 --> 00:41:30,780
and understand exactly,
is trapped between log N

809
00:41:30,780 --> 00:41:34,260
and log N + 1.

810
00:41:34,260 --> 00:41:35,160
Yeah question.

811
00:41:35,160 --> 00:41:36,326
AUDIENCE: [INAUDIBLE PHRASE]

812
00:41:36,326 --> 00:41:46,039


813
00:41:46,039 --> 00:41:47,580
PROFESSOR: This step
here is the step

814
00:41:47,580 --> 00:41:49,550
that you're concerned about.

815
00:41:49,550 --> 00:41:54,160
So this step is a
geometric argument which

816
00:41:54,160 --> 00:41:57,070
is analogous to this step.

817
00:41:57,070 --> 00:42:01,320
All right it's the
same type of argument.

818
00:42:01,320 --> 00:42:04,720
And in this case it's that
the rectangles are on top

819
00:42:04,720 --> 00:42:07,270
and so the area represented
on the right hand side

820
00:42:07,270 --> 00:42:09,840
is less than the area
represented on this side.

821
00:42:09,840 --> 00:42:11,560
And this is the
same type of thing

822
00:42:11,560 --> 00:42:14,400
except that the
rectangles are underneath.

823
00:42:14,400 --> 00:42:17,310
So the sum of the
areas of the rectangles

824
00:42:17,310 --> 00:42:19,190
is less than the
area under the curve.

825
00:42:19,190 --> 00:42:23,741


826
00:42:23,741 --> 00:42:24,240
All right.

827
00:42:24,240 --> 00:42:27,380
So I've now trapped
this quantity.

828
00:42:27,380 --> 00:42:34,440
And I'm now going to state
the sort of general results.

829
00:42:34,440 --> 00:42:38,420


830
00:42:38,420 --> 00:42:42,230
So here's what's known
as integral comparison.

831
00:42:42,230 --> 00:42:44,900
It's this double
arrow correspondence

832
00:42:44,900 --> 00:42:54,580
in the general case,
for a very general case.

833
00:42:54,580 --> 00:42:57,130
There are actually even
more cases where it works.

834
00:42:57,130 --> 00:43:01,220
But this is a good
case and convenient.

835
00:43:01,220 --> 00:43:02,845
Now this is called
integral comparison.

836
00:43:02,845 --> 00:43:07,060


837
00:43:07,060 --> 00:43:12,030
And it comes with hypotheses
but it follows the same argument

838
00:43:12,030 --> 00:43:13,495
that I just gave.

839
00:43:13,495 --> 00:43:27,540
If f(x) is decreasing
and it's positive,

840
00:43:27,540 --> 00:43:37,520
then the sum f(n), n
equals 1 to infinity,

841
00:43:37,520 --> 00:43:44,570
minus the integral from
1 to infinity of f(x) dx

842
00:43:44,570 --> 00:43:45,420
is less than f(1).

843
00:43:45,420 --> 00:43:50,060


844
00:43:50,060 --> 00:43:51,590
That's basically what we showed.

845
00:43:51,590 --> 00:43:54,570
We showed that the difference
between S_N and log N

846
00:43:54,570 --> 00:43:55,930
was at most 1.

847
00:43:55,930 --> 00:43:59,321


848
00:43:59,321 --> 00:43:59,820
All right.

849
00:43:59,820 --> 00:44:11,590
Now if both of them
are-- And the sum

850
00:44:11,590 --> 00:44:25,380
and the integral converge
or diverge together.

851
00:44:25,380 --> 00:44:27,750
That is they either both
converge or both diverge.

852
00:44:27,750 --> 00:44:30,780
This is the type of test that
we like because then we can just

853
00:44:30,780 --> 00:44:33,000
convert the question of
convergence over here

854
00:44:33,000 --> 00:44:37,770
to this question of convergence
over on the other side.

855
00:44:37,770 --> 00:44:41,590
Now I remind you that
it's incredibly hard

856
00:44:41,590 --> 00:44:44,870
to calculate these numbers.

857
00:44:44,870 --> 00:44:47,220
Whereas these numbers
are easier to calculate.

858
00:44:47,220 --> 00:44:50,390
Our goal is to reduce
things to simpler things.

859
00:44:50,390 --> 00:44:52,910
And in this case
sums, infinite sums

860
00:44:52,910 --> 00:44:54,720
are much harder than
infinite integrals.

861
00:44:54,720 --> 00:45:00,080


862
00:45:00,080 --> 00:45:03,150
All right so that's the
integral comparison.

863
00:45:03,150 --> 00:45:12,140
And now I have one
last bit on comparisons

864
00:45:12,140 --> 00:45:13,840
that I need to tell you about.

865
00:45:13,840 --> 00:45:16,310
And this is very much like
what we did with integrals.

866
00:45:16,310 --> 00:45:18,240
Which is a so called
limit comparison.

867
00:45:18,240 --> 00:45:29,200


868
00:45:29,200 --> 00:45:31,190
The limit comparison
says the following:

869
00:45:31,190 --> 00:45:45,210
if f(n) is similar to g(n) --
recall that means f(n) / g(n)

870
00:45:45,210 --> 00:45:51,760
tends to 1 as n
goes to infinity --

871
00:45:51,760 --> 00:45:55,000
and we're in the positive case.

872
00:45:55,000 --> 00:45:57,535
So let's just say
g(n) is positive.

873
00:45:57,535 --> 00:46:03,920


874
00:46:03,920 --> 00:46:05,490
Then-- that doesn't
even, well-- then

875
00:46:05,490 --> 00:46:15,055
sum f(n), sum g(n)
either both-- same thing

876
00:46:15,055 --> 00:46:21,884
as above, either both
converge or both diverge.

877
00:46:21,884 --> 00:46:27,488


878
00:46:27,488 --> 00:46:28,770
All right.

879
00:46:28,770 --> 00:46:30,730
This is just saying
that if they behave

880
00:46:30,730 --> 00:46:33,920
the same way in the tail, which
is all we really care about,

881
00:46:33,920 --> 00:46:40,820
then they have similar behavior,
similar convergence properties.

882
00:46:40,820 --> 00:46:44,107


883
00:46:44,107 --> 00:46:45,690
And let me give you
a couple examples.

884
00:46:45,690 --> 00:46:50,370


885
00:46:50,370 --> 00:46:56,580
So here's one example: if you
take the sum 1 over n^2 + 1,

886
00:46:56,580 --> 00:46:57,080
square root.

887
00:46:57,080 --> 00:47:01,890


888
00:47:01,890 --> 00:47:05,350
This is going to be replaced
by something simpler.

889
00:47:05,350 --> 00:47:07,240
Which is the main term here.

890
00:47:07,240 --> 00:47:10,460
Which is 1 over
square root of n^2,

891
00:47:10,460 --> 00:47:15,100
which we recognize as
sum 1/n, which diverges.

892
00:47:15,100 --> 00:47:17,920


893
00:47:17,920 --> 00:47:20,440
So this guy is one
of the red guys.

894
00:47:20,440 --> 00:47:24,300


895
00:47:24,300 --> 00:47:26,410
On the red team.

896
00:47:26,410 --> 00:47:30,330
Now we have another example.

897
00:47:30,330 --> 00:47:33,370


898
00:47:33,370 --> 00:47:38,960
Which is let's say the square
root of n, I don't know,

899
00:47:38,960 --> 00:47:43,080
to the fifth minus n^2.

900
00:47:43,080 --> 00:47:44,665
Now if you have
something where it's

901
00:47:44,665 --> 00:47:46,290
negative in the
denominator you kind of

902
00:47:46,290 --> 00:47:49,540
do have to watch out that
denominator makes sense.

903
00:47:49,540 --> 00:47:50,570
It isn't 0.

904
00:47:50,570 --> 00:47:53,042
So we're going to be careful
and start this at n = 2.

905
00:47:53,042 --> 00:48:00,370
In which case, the first
term, I don't like 1/0

906
00:48:00,370 --> 00:48:01,720
as a term in my series.

907
00:48:01,720 --> 00:48:04,130
So I'm just going to be a
little careful about how--

908
00:48:04,130 --> 00:48:05,710
as I said I was
kind of lazy here.

909
00:48:05,710 --> 00:48:09,930
I could have started this
one at 0 for instance.

910
00:48:09,930 --> 00:48:10,810
All right.

911
00:48:10,810 --> 00:48:14,110
So here's the picture.

912
00:48:14,110 --> 00:48:19,070
Now this I just replace by its
main term which is 1 over n^5,

913
00:48:19,070 --> 00:48:20,340
square root.

914
00:48:20,340 --> 00:48:25,975
Which is sum 1/n^(5/2),
which converges.

915
00:48:25,975 --> 00:48:28,705


916
00:48:28,705 --> 00:48:29,520
All right.

917
00:48:29,520 --> 00:48:30,800
The power is bigger than 1.

918
00:48:30,800 --> 00:48:33,960
1 is the divider for these
things and it just misses.

919
00:48:33,960 --> 00:48:38,752
This one converges.

920
00:48:38,752 --> 00:48:41,800
All right so these
are the typical ways

921
00:48:41,800 --> 00:48:47,355
in which these convergence
processes are used.

922
00:48:47,355 --> 00:48:47,870
All right.

923
00:48:47,870 --> 00:48:49,930
So I have one more
thing for you.

924
00:48:49,930 --> 00:48:52,600
Which is an advertisement
for next time.

925
00:48:52,600 --> 00:48:56,360
And I have this demo
here which I will grab.

926
00:48:56,360 --> 00:48:58,300
But you will see this next time.

927
00:48:58,300 --> 00:49:00,880
So here's a question for
you to think about overnight

928
00:49:00,880 --> 00:49:04,240
but don't ask friends, you have
to think about it yourself.

929
00:49:04,240 --> 00:49:05,492
So here's the problem.

930
00:49:05,492 --> 00:49:08,000
Here are some blocks
which I acquired

931
00:49:08,000 --> 00:49:09,390
when my kids left home.

932
00:49:09,390 --> 00:49:12,190


933
00:49:12,190 --> 00:49:20,550
Anyway yeah that'll happen to
you too in about four years.

934
00:49:20,550 --> 00:49:25,929
So now here you are,
these are blocks.

935
00:49:25,929 --> 00:49:27,470
So now here's the
question that we're

936
00:49:27,470 --> 00:49:29,546
going to deal with next time.

937
00:49:29,546 --> 00:49:30,920
I'm going to build
it, maybe I'll

938
00:49:30,920 --> 00:49:32,961
put it over here because
I want to have some room

939
00:49:32,961 --> 00:49:34,890
to head this way.

940
00:49:34,890 --> 00:49:41,320
I want to stack them up
so that-- oh didn't work.

941
00:49:41,320 --> 00:49:44,430
Going to stack them up
in the following way.

942
00:49:44,430 --> 00:49:48,280
I want to do it so that
the top one is completely

943
00:49:48,280 --> 00:49:51,540
to the right of the bottom one.

944
00:49:51,540 --> 00:49:53,170
That's the question
can I do that?

945
00:49:53,170 --> 00:49:57,870
Can I get-- Can I build this up?

946
00:49:57,870 --> 00:50:01,980
So let's see here.

947
00:50:01,980 --> 00:50:04,590
I just seem to be missing--
but anyway what I'm going to do

948
00:50:04,590 --> 00:50:05,965
is I'm going to
try to build this

949
00:50:05,965 --> 00:50:10,880
and we're going to see how far
we can get with this next time.

950
00:50:10,880 --> 00:50:11,678