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HERBERT GROSS: Hi, last time I
left you dangling in suspense
11
00:00:35,290 --> 00:00:38,830
when I said what happens if the
terms in a series are not
12
00:00:38,830 --> 00:00:39,580
all positive?
13
00:00:39,580 --> 00:00:42,550
In other words, the trouble with
the last assignment was
14
00:00:42,550 --> 00:00:45,100
that we did quite a bit of work
and yet there was a very
15
00:00:45,100 --> 00:00:46,080
stringent condition.
16
00:00:46,080 --> 00:00:49,770
Namely, that every term that you
were adding happened to be
17
00:00:49,770 --> 00:00:50,790
a positive number.
18
00:00:50,790 --> 00:00:53,320
Now obviously, this need
not be the case.
19
00:00:53,320 --> 00:00:57,750
And the question is, can you
have convergence in a series
20
00:00:57,750 --> 00:01:00,300
in which the terms are
not all positive?
21
00:01:00,300 --> 00:01:03,100
And what does it mean, by the
way, if this is the case why
22
00:01:03,100 --> 00:01:06,980
we stressed the situation
of positive series?
23
00:01:06,980 --> 00:01:09,430
And this will be the aim of
today's lecture to straighten
24
00:01:09,430 --> 00:01:10,705
out both of these points.
25
00:01:10,705 --> 00:01:14,430
At any rate, I call today's
lesson 'Absolute Convergence',
26
00:01:14,430 --> 00:01:17,450
and I hope that the meaning of
this will become clear very
27
00:01:17,450 --> 00:01:18,750
soon as we go along.
28
00:01:18,750 --> 00:01:21,810
But to answer the first question
that we brought up,
29
00:01:21,810 --> 00:01:24,110
let's take a series
in which the
30
00:01:24,110 --> 00:01:25,950
terms are not all positive.
31
00:01:25,950 --> 00:01:29,450
Now, by the way, I will do more
general things in our
32
00:01:29,450 --> 00:01:31,470
supplementary notes
on this material.
33
00:01:31,470 --> 00:01:34,170
I felt though that for a
blackboard illustration, I
34
00:01:34,170 --> 00:01:37,110
should pick a relatively
straightforward example and
35
00:01:37,110 --> 00:01:38,720
not try to generalize it.
36
00:01:38,720 --> 00:01:46,140
Let's take the specific series 1
minus 1/2 plus 1/3 minus 1/4
37
00:01:46,140 --> 00:01:47,750
plus 1/5, et cetera.
38
00:01:47,750 --> 00:01:50,470
And how do we indicate the
n-th term in this case?
39
00:01:50,470 --> 00:01:53,400
Notice that the denominator
is 'n'.
40
00:01:53,400 --> 00:01:57,840
The numerator oscillates
between minus 1 and 1.
41
00:01:57,840 --> 00:02:01,720
And you see the mathematical
trick to alternate signs is to
42
00:02:01,720 --> 00:02:03,690
raise minus 1 to a power.
43
00:02:03,690 --> 00:02:07,290
You see minus 1 to an even
power will be positive 1.
44
00:02:07,290 --> 00:02:09,210
And to an odd power,
negative 1.
45
00:02:09,210 --> 00:02:12,860
The idea here is since for 'n'
equals 1 we want this to be
46
00:02:12,860 --> 00:02:15,370
positive, we tacked on
the plus 1 here.
47
00:02:15,370 --> 00:02:19,780
You see in that case, 'n' being
1, 'n plus 1' is 2.
48
00:02:19,780 --> 00:02:21,840
Minus 1 squared is 1.
49
00:02:21,840 --> 00:02:25,320
And at any rate, don't be
confused by this notation,
50
00:02:25,320 --> 00:02:28,700
it's simply a cute way all
of alternating signs.
51
00:02:28,700 --> 00:02:31,560
At any rate, what do we have
in this particular series?
52
00:02:31,560 --> 00:02:35,530
We have that the terms
alternate in sign.
53
00:02:35,530 --> 00:02:39,050
We also have that the n-th
term approaches 0.
54
00:02:39,050 --> 00:02:41,750
Namely, the numerator
alternates between
55
00:02:41,750 --> 00:02:43,220
plus and minus 1.
56
00:02:43,220 --> 00:02:44,990
The denominator is 'n'.
57
00:02:44,990 --> 00:02:50,710
So as 'n' increases, the terms
converge on 0 as a limit.
58
00:02:50,710 --> 00:02:54,470
And finally, the terms
keep decreasing
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00:02:54,470 --> 00:02:56,650
monotonically in magnitude.
60
00:02:56,650 --> 00:02:59,620
In other words, forgetting about
the fact that a plus
61
00:02:59,620 --> 00:03:03,310
outranks a minus in terms of a
number line, notice that the
62
00:03:03,310 --> 00:03:06,930
size of 1/3 is less than
the size of 1/2.
63
00:03:06,930 --> 00:03:08,990
In other words, this particular
series, which is
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00:03:08,990 --> 00:03:12,600
called an 'alternating series',
has three properties.
65
00:03:12,600 --> 00:03:14,680
It's called alternating
because the terms
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00:03:14,680 --> 00:03:16,450
alternate in sign.
67
00:03:16,450 --> 00:03:18,080
The n-th term approaches 0.
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00:03:18,080 --> 00:03:21,600
By the way, we saw in our first
lecture on series that
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00:03:21,600 --> 00:03:24,980
the n-th term approaching
0 was necessary, but not
70
00:03:24,980 --> 00:03:27,850
sufficient for making
a series converge.
71
00:03:27,850 --> 00:03:32,120
However, what our claim is that
if in addition to this we
72
00:03:32,120 --> 00:03:35,140
know that the terms decrease
in magnitude, our claim is
73
00:03:35,140 --> 00:03:37,460
that the given series
will converge.
74
00:03:37,460 --> 00:03:40,770
In other words, what I intend
to show is that 1 minus 1/2
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00:03:40,770 --> 00:03:43,670
plus 1/3, et cetera,
does converge.
76
00:03:43,670 --> 00:03:46,580
And the way I'm going to do
that is geometrically.
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00:03:46,580 --> 00:03:49,320
I'm not going to try to prove
this thing analytically.
78
00:03:49,320 --> 00:03:53,740
But as I've said before, the
analytic proof virtually is
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00:03:53,740 --> 00:03:56,990
just an abstraction of what
we're doing over here.
80
00:03:56,990 --> 00:03:59,650
Let's take a look and see
what happens over here.
81
00:03:59,650 --> 00:04:03,105
Notice that our first
term is 1.
82
00:04:03,105 --> 00:04:06,005
Our next term is 1/2.
83
00:04:06,005 --> 00:04:10,670
Our next term is 1 minus 1/2
plus 1/3, which is 5/6.
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00:04:10,670 --> 00:04:14,730
What I'm driving at is that if
you compute the sums, 1 minus
85
00:04:14,730 --> 00:04:17,970
1/2 plus 1/3 minus 1/4, et
cetera, in the order in which
86
00:04:17,970 --> 00:04:20,630
they're given, what you find
is that the sequence of
87
00:04:20,630 --> 00:04:30,040
partial sums is 1, 1/2, 5/6,
7/12, 47/60, 13/20.
88
00:04:30,040 --> 00:04:32,660
Now the thing that I want you to
see, the reason I'm waving
89
00:04:32,660 --> 00:04:35,700
my hand here and why I think
this helps in the lecture is
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00:04:35,700 --> 00:04:36,870
look what's happening.
91
00:04:36,870 --> 00:04:40,190
You see because the terms
alternate in sign what this
92
00:04:40,190 --> 00:04:46,640
means is that as I start with 's
sub 1' over here, the next
93
00:04:46,640 --> 00:04:49,600
term will be to the
left of 's sub 1'.
94
00:04:49,600 --> 00:04:52,120
Then the term after that
will be to the right.
95
00:04:52,120 --> 00:04:53,440
Then to the left.
96
00:04:53,440 --> 00:04:54,550
Then to the right.
97
00:04:54,550 --> 00:04:55,590
Then to the left.
98
00:04:55,590 --> 00:04:56,460
Then to the right.
99
00:04:56,460 --> 00:04:58,230
They keep alternating
this way.
100
00:04:58,230 --> 00:05:03,500
Moreover, since the terms
decrease in magnitude, it
101
00:05:03,500 --> 00:05:06,750
means that each jump is less
than the jump that came
102
00:05:06,750 --> 00:05:08,060
immediately before.
103
00:05:08,060 --> 00:05:10,910
In other words, as I jumped from
here to here, when I jump
104
00:05:10,910 --> 00:05:13,310
back I don't come back
quite as far.
105
00:05:13,310 --> 00:05:16,060
In other words, what I'm doing
now is I'm closing in.
106
00:05:16,060 --> 00:05:20,140
You see the odd subscripts and
the even subscripts are sort
107
00:05:20,140 --> 00:05:21,400
of segregated.
108
00:05:21,400 --> 00:05:22,960
You see what's happening
over here?
109
00:05:22,960 --> 00:05:28,590
And finally, because the limit
of the n-th term is 0, it
110
00:05:28,590 --> 00:05:30,370
means that this spacing--
111
00:05:30,370 --> 00:05:32,420
see the difference between
successive partial
112
00:05:32,420 --> 00:05:33,920
sums is the n-th term.
113
00:05:33,920 --> 00:05:35,380
That must go to 0.
114
00:05:35,380 --> 00:05:38,000
In other words, our limit
'L' is in here.
115
00:05:38,000 --> 00:05:42,120
And as 'n' increases, the
squeeze is put on and we get
116
00:05:42,120 --> 00:05:43,280
the existence of a limit.
117
00:05:43,280 --> 00:05:45,720
For example, whatever
'L' is, it must be
118
00:05:45,720 --> 00:05:49,500
between 13/20 and 47/60.
119
00:05:49,500 --> 00:05:51,910
By the way, I say this in
the form of an aside.
120
00:05:51,910 --> 00:05:53,270
It turns out--
121
00:05:53,270 --> 00:05:56,130
and for those of us who haven't
seen this before, it's
122
00:05:56,130 --> 00:05:57,330
a very mystic result.
123
00:05:57,330 --> 00:05:58,990
That's why I say, who'd
have guessed it?
124
00:05:58,990 --> 00:06:02,420
It turns out that 'L' is
actually the natural log of 2.
125
00:06:02,420 --> 00:06:05,380
And the reason I point this
out is that again, you may
126
00:06:05,380 --> 00:06:08,090
recall that in our notes
we talked about 'Cauchy
127
00:06:08,090 --> 00:06:11,540
convergence', meaning what do
you do when you don't know how
128
00:06:11,540 --> 00:06:12,740
to guess the limit?
129
00:06:12,740 --> 00:06:16,180
You see, the idea here is notice
that this particular
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00:06:16,180 --> 00:06:18,250
series converges.
131
00:06:18,250 --> 00:06:20,820
But we don't know what the limit
is other than the fact
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00:06:20,820 --> 00:06:22,890
that it's being squeezed
in over here.
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00:06:22,890 --> 00:06:25,840
You see here's a case where we
know that a limit exists, but
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00:06:25,840 --> 00:06:28,740
it's particularly difficult
to explicitly name
135
00:06:28,740 --> 00:06:30,010
what that limit is.
136
00:06:30,010 --> 00:06:34,120
But that fact notwithstanding,
is it clear that because of
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00:06:34,120 --> 00:06:36,930
the fact that the terms
alternate, the magnitudes
138
00:06:36,930 --> 00:06:40,530
decrease, and the limit is 0
that these things do converge
139
00:06:40,530 --> 00:06:41,960
to a limit?
140
00:06:41,960 --> 00:06:43,350
I think it is clear.
141
00:06:43,350 --> 00:06:46,150
But the next question, and I
apologize for what looks like
142
00:06:46,150 --> 00:06:49,440
slang here, but I think this is
exactly what's going on in
143
00:06:49,440 --> 00:06:50,460
your minds right now.
144
00:06:50,460 --> 00:06:52,620
So what?
145
00:06:52,620 --> 00:06:55,880
What does this have to do with
what came before and what will
146
00:06:55,880 --> 00:06:56,780
come later?
147
00:06:56,780 --> 00:07:00,480
And we're going to see again,
a very, very strange thing
148
00:07:00,480 --> 00:07:04,120
that happens with infinite sums
that does not happen with
149
00:07:04,120 --> 00:07:05,310
finite sums.
150
00:07:05,310 --> 00:07:08,930
Let me lead into that
fairly gradually.
151
00:07:08,930 --> 00:07:12,800
First of all, I claim that
this particular series--
152
00:07:12,800 --> 00:07:14,740
see, again, don't get
misled by this.
153
00:07:14,740 --> 00:07:16,230
It's just a fancy way
of saying what?
154
00:07:16,230 --> 00:07:21,230
1 minus 1/2 plus 1/3, et
cetera, converges.
155
00:07:21,230 --> 00:07:24,770
But because the plus terms
cancel the minus terms, the
156
00:07:24,770 --> 00:07:29,740
pluses cancel the minuses, not
because the terms get small
157
00:07:29,740 --> 00:07:30,660
fast enough.
158
00:07:30,660 --> 00:07:32,670
What I mean by that is this.
159
00:07:32,670 --> 00:07:34,390
Forget about the
signs in here.
160
00:07:34,390 --> 00:07:36,790
Replace each term by
its magnitude.
161
00:07:36,790 --> 00:07:38,990
And by the way, that's where
the name 'absolute'
162
00:07:38,990 --> 00:07:40,980
convergence is going
to come from.
163
00:07:40,980 --> 00:07:45,060
Namely, the magnitude of a term
is its absolute value.
164
00:07:45,060 --> 00:07:46,640
And this is what we're going to
be talking about, but the
165
00:07:46,640 --> 00:07:47,520
idea is this.
166
00:07:47,520 --> 00:07:50,760
If we replace each term
by its magnitude,
167
00:07:50,760 --> 00:07:52,500
we obtain the series.
168
00:07:52,500 --> 00:07:55,810
Summation 'n' goes from 1
to infinity '1 over n'.
169
00:07:55,810 --> 00:08:00,450
And in the last assignment, we
saw in the exercises that this
170
00:08:00,450 --> 00:08:03,000
diverged by the integral test.
171
00:08:03,000 --> 00:08:06,400
In other words, if we leave
out the signs, the series
172
00:08:06,400 --> 00:08:09,060
diverges because evidently these
terms don't get small
173
00:08:09,060 --> 00:08:09,575
fast enough.
174
00:08:09,575 --> 00:08:10,750
Okay.
175
00:08:10,750 --> 00:08:13,160
Let me state a definition.
176
00:08:13,160 --> 00:08:14,620
The definition is simply this.
177
00:08:14,620 --> 00:08:17,140
178
00:08:17,140 --> 00:08:20,720
The series, summation 'n' goes
from 1 to infinity, 'a sub n'
179
00:08:20,720 --> 00:08:28,300
is said to converge absolutely
if the series that you get by
180
00:08:28,300 --> 00:08:33,179
replacing each term by its
magnitude converges.
181
00:08:33,179 --> 00:08:36,159
Now I leave for the
supplementary notes the proof
182
00:08:36,159 --> 00:08:40,130
that if a series converges
absolutely, it converges in
183
00:08:40,130 --> 00:08:40,750
the first place.
184
00:08:40,750 --> 00:08:42,970
In other words, I think it's
rather clear if you look at
185
00:08:42,970 --> 00:08:45,950
this thing intuitively that if
I replace each term by its
186
00:08:45,950 --> 00:08:48,660
magnitude, I'll disregard the
plus and minus signs.
187
00:08:48,660 --> 00:08:51,780
And that resulting series
converges, then the original
188
00:08:51,780 --> 00:08:54,910
series must've converged also
because the terms couldn't be
189
00:08:54,910 --> 00:08:56,380
any bigger than this.
190
00:08:56,380 --> 00:08:59,600
By the way, the formal proof is
kind of messy in places and
191
00:08:59,600 --> 00:09:01,570
so this is why, as I say,
I leave this for the
192
00:09:01,570 --> 00:09:03,860
supplementary notes.
193
00:09:03,860 --> 00:09:07,040
But at any rate that's what we
mean by absolute convergence.
194
00:09:07,040 --> 00:09:09,380
If you're given a series,
you replace each
195
00:09:09,380 --> 00:09:11,190
term by it's magnitude.
196
00:09:11,190 --> 00:09:15,250
If that series converges, we
call the original series
197
00:09:15,250 --> 00:09:16,830
absolutely convergent.
198
00:09:16,830 --> 00:09:21,160
Notice by the way, the tie
in now between absolute
199
00:09:21,160 --> 00:09:23,120
convergence and positive
series.
200
00:09:23,120 --> 00:09:26,610
Namely, by definition, the
absolute value of 'a sub n' is
201
00:09:26,610 --> 00:09:27,940
at least as big as 0.
202
00:09:27,940 --> 00:09:30,420
Consequently, when we're
testing for absolute
203
00:09:30,420 --> 00:09:33,880
convergence, the series that
we test is positive.
204
00:09:33,880 --> 00:09:36,620
And we have tests
for convergence
205
00:09:36,620 --> 00:09:38,460
for positive series.
206
00:09:38,460 --> 00:09:41,410
Now the sequel to definition
one is of course definition
207
00:09:41,410 --> 00:09:42,990
two, and that says what?
208
00:09:42,990 --> 00:09:47,040
A series which converges but
not absolutely is called
209
00:09:47,040 --> 00:09:48,610
conditionally convergent.
210
00:09:48,610 --> 00:09:52,540
In other words, it converges on
the condition that the sine
211
00:09:52,540 --> 00:09:54,840
stay exactly the way they are.
212
00:09:54,840 --> 00:09:58,340
An example of a conditionally
convergent series is the one
213
00:09:58,340 --> 00:10:00,020
that we're dealing
with right now.
214
00:10:00,020 --> 00:10:02,680
Namely, with the pluses and
minuses in there, we just
215
00:10:02,680 --> 00:10:04,440
showed that the series
converges.
216
00:10:04,440 --> 00:10:08,000
However, if we replace each
term by its magnitude, the
217
00:10:08,000 --> 00:10:10,990
resulting series is summation
'1 over n'.
218
00:10:10,990 --> 00:10:14,230
And that as we saw, diverges.
219
00:10:14,230 --> 00:10:17,740
Now the question is, what's
so bad about conditional
220
00:10:17,740 --> 00:10:18,840
convergence?
221
00:10:18,840 --> 00:10:21,490
What difference does it make
whether a series converges
222
00:10:21,490 --> 00:10:23,530
absolutely or conditionally?
223
00:10:23,530 --> 00:10:25,790
Is there any problem
that comes up?
224
00:10:25,790 --> 00:10:29,370
As I said before, a fantastic
subtlety that occurs, a
225
00:10:29,370 --> 00:10:32,610
subtlety that has no parallel
in our knowledge of finite
226
00:10:32,610 --> 00:10:33,710
arithmetic.
227
00:10:33,710 --> 00:10:35,650
The subtlety is this.
228
00:10:35,650 --> 00:10:37,470
In fact, I call it that, the
subtlety of conditional
229
00:10:37,470 --> 00:10:38,500
convergence.
230
00:10:38,500 --> 00:10:40,570
Namely, the sum of a
231
00:10:40,570 --> 00:10:42,810
conditionally convergent series--
232
00:10:42,810 --> 00:10:45,200
and this fantastic--
233
00:10:45,200 --> 00:10:49,440
depends on the order in which
you write the terms.
234
00:10:49,440 --> 00:10:53,870
In other words, if the series
converges, but conditionally
235
00:10:53,870 --> 00:10:57,680
if you change the order of the
terms, surprising as it may
236
00:10:57,680 --> 00:11:00,970
seem, you actually
change the sum.
237
00:11:00,970 --> 00:11:03,820
And the best way to do this
I think at this stage, a
238
00:11:03,820 --> 00:11:07,060
generalization is given in
the supplementary notes.
239
00:11:07,060 --> 00:11:09,800
But the idea is, let me just
do this in terms of the
240
00:11:09,800 --> 00:11:11,380
problem that we were
dealing with.
241
00:11:11,380 --> 00:11:13,820
Let's take the terms of our
conditionally convergent
242
00:11:13,820 --> 00:11:17,570
series, 1 minus 1/2 plus 1/3,
et cetera, and divide them
243
00:11:17,570 --> 00:11:19,180
into two teams.
244
00:11:19,180 --> 00:11:22,700
And if this expression bothers
you, call the teams sets and
245
00:11:22,700 --> 00:11:25,000
that makes it much more
mathematical.
246
00:11:25,000 --> 00:11:27,880
Let the first set
consist of the
247
00:11:27,880 --> 00:11:29,310
positive terms of a series.
248
00:11:29,310 --> 00:11:32,320
Namely, 1, 1/3, 1/5.
249
00:11:32,320 --> 00:11:36,450
And in general, '1 over '2n
minus 1'' where 'n' is any
250
00:11:36,450 --> 00:11:38,500
positive whole number.
251
00:11:38,500 --> 00:11:42,030
The negative numbers of our
team are minus 1/2--
252
00:11:42,030 --> 00:11:43,950
or the set 'N', which I'll call
the negative members is
253
00:11:43,950 --> 00:11:46,550
minus 1/2, minus
1/4, minus 1/6.
254
00:11:46,550 --> 00:11:49,060
In general, minus '1 over 2n'.
255
00:11:49,060 --> 00:11:52,980
Now again, by the integral
test, we saw in our last
256
00:11:52,980 --> 00:11:57,700
lesson that both these series
summation '1 all over '2n
257
00:11:57,700 --> 00:12:02,500
minus 1'' and summation '1 over
2n' diverge to infinity
258
00:12:02,500 --> 00:12:04,430
by the integral test.
259
00:12:04,430 --> 00:12:07,140
Now what my claim is, is that
because both of these two
260
00:12:07,140 --> 00:12:11,870
series diverge, I can now
rearrange my terms to get any
261
00:12:11,870 --> 00:12:13,150
sum that I want.
262
00:12:13,150 --> 00:12:16,540
Well, for example, suppose
somebody says to me, make the
263
00:12:16,540 --> 00:12:19,960
sum come out to be 3/2.
264
00:12:19,960 --> 00:12:22,170
As I'll mention later, there's
nothing sacred about 3/2.
265
00:12:22,170 --> 00:12:24,680
In fact, I'm mentioning that
now but I'll say it again
266
00:12:24,680 --> 00:12:25,860
later for emphasis.
267
00:12:25,860 --> 00:12:29,300
I just wanted to pick a number
that wouldn't be too unwieldy.
268
00:12:29,300 --> 00:12:30,610
But here's what the gist is.
269
00:12:30,610 --> 00:12:32,850
I want to make the
sum at least 3/2.
270
00:12:32,850 --> 00:12:36,050
So what I do is I start with my
positive terms from the set
271
00:12:36,050 --> 00:12:37,500
'P' and add them up.
272
00:12:37,500 --> 00:12:41,360
1 plus 1/3 plus 1/5,
et cetera.
273
00:12:41,360 --> 00:12:44,700
The thing that I'm sure of is
that eventually this sum must
274
00:12:44,700 --> 00:12:46,120
exceed 3/2.
275
00:12:46,120 --> 00:12:47,560
Why do I know that?
276
00:12:47,560 --> 00:12:50,270
Well, 'P' diverges
to infinity.
277
00:12:50,270 --> 00:12:54,400
How could 'P' possibly diverge
to infinity if the sum could
278
00:12:54,400 --> 00:12:57,730
never-- the sum of the terms in
'P' diverges to infinity.
279
00:12:57,730 --> 00:12:59,350
How could that happen if
the sum never got at
280
00:12:59,350 --> 00:13:01,630
least as big as 3/2?
281
00:13:01,630 --> 00:13:05,850
So the idea is I write down all
of these terms, add them
282
00:13:05,850 --> 00:13:09,810
up, until my sum first exceeds
or equals 3/2.
283
00:13:09,810 --> 00:13:12,740
And as I say, this must
happen because the
284
00:13:12,740 --> 00:13:14,560
series diverges to infinity.
285
00:13:14,560 --> 00:13:17,790
Well, in particular, I
observe that 1 plus
286
00:13:17,790 --> 00:13:21,110
1/3 plus 1/5 is 23/15.
287
00:13:21,110 --> 00:13:23,970
And that's now, for the first
time, bigger than 3/2.
288
00:13:23,970 --> 00:13:27,460
What I do next is I annex the
negative members, in other
289
00:13:27,460 --> 00:13:29,810
words, the members of capital
'N', until the
290
00:13:29,810 --> 00:13:32,540
sum falls below 3/2.
291
00:13:32,540 --> 00:13:35,580
Watch what I'm doing here.
292
00:13:35,580 --> 00:13:36,930
I stopped at 1/5.
293
00:13:36,930 --> 00:13:39,010
I have 1 plus 1/3 plus 1/5.
294
00:13:39,010 --> 00:13:40,710
Now I subtract 1/2.
295
00:13:40,710 --> 00:13:44,710
That gives me 31/30 and
that's less than 3/2.
296
00:13:44,710 --> 00:13:46,810
So now my sum is below 3/2.
297
00:13:46,810 --> 00:13:51,250
What I do next is I continue
with 'P' where I left off.
298
00:13:51,250 --> 00:13:53,130
See I left off with 1/5.
299
00:13:53,130 --> 00:14:00,420
I now start tacking on 1/7, 1/9,
1/11, 1/13, et cetera.
300
00:14:00,420 --> 00:14:04,380
Until the sum, again,
exceeds 3/2.
301
00:14:04,380 --> 00:14:07,050
Now how do I know that the
sum has to exceed 3/2?
302
00:14:07,050 --> 00:14:11,520
Well, remember when we add up
all of the members of 'P', we
303
00:14:11,520 --> 00:14:13,170
get a divergent series.
304
00:14:13,170 --> 00:14:16,980
That means the sum increases
without bound.
305
00:14:16,980 --> 00:14:20,120
As we've mentioned many times so
far in our course, that if
306
00:14:20,120 --> 00:14:24,430
you chop off a finite number
of terms from a divergent
307
00:14:24,430 --> 00:14:26,830
series, the remaining
series, what's
308
00:14:26,830 --> 00:14:28,920
left, still must diverge.
309
00:14:28,920 --> 00:14:34,550
In other words, if this series
1 plus 1/3 plus 1/5 plus 1/7
310
00:14:34,550 --> 00:14:38,190
diverges to infinity, the fact
that I chop off those terms
311
00:14:38,190 --> 00:14:41,740
that add up to just an excess
of 3/2, what's left is still
312
00:14:41,740 --> 00:14:43,380
going to diverge to infinity.
313
00:14:43,380 --> 00:14:45,520
So I can keep on
going this way.
314
00:14:45,520 --> 00:14:47,450
What I do is I add on 1/7.
315
00:14:47,450 --> 00:14:50,830
The result turns out to
be 247/210, which is
316
00:14:50,830 --> 00:14:52,320
still less than 3/2.
317
00:14:52,320 --> 00:14:54,080
And to spare you the
gory details--
318
00:14:54,080 --> 00:14:55,690
and believe me, they are gory.
319
00:14:55,690 --> 00:14:57,320
I worked it out myself.
320
00:14:57,320 --> 00:15:00,030
Without a desk calculator
this gets to be a mess.
321
00:15:00,030 --> 00:15:03,480
It turns out that when I add on
1/13, get down to here, the
322
00:15:03,480 --> 00:15:06,570
sum is this, which is
still less than 3/2.
323
00:15:06,570 --> 00:15:08,600
But then I add on 1/15.
324
00:15:08,600 --> 00:15:11,640
The sum gets to be this, which
I simply call 'k'.
325
00:15:11,640 --> 00:15:14,360
That turns out to be
greater than 3/2.
326
00:15:14,360 --> 00:15:17,330
Then you see what I do is
return to my series of
327
00:15:17,330 --> 00:15:21,140
negative terms, tack those on
till the sum falls below.
328
00:15:21,140 --> 00:15:24,970
And what's happening here
pictorially is the following.
329
00:15:24,970 --> 00:15:27,660
You see what happened
was 3/2--
330
00:15:27,660 --> 00:15:31,290
I added on terms till
I exceeded 3/2.
331
00:15:31,290 --> 00:15:33,730
That was 23/15.
332
00:15:33,730 --> 00:15:37,580
Then I get down below 3/2.
333
00:15:37,580 --> 00:15:39,760
Then up again above 3/2.
334
00:15:39,760 --> 00:15:41,860
And still sparing
you the details,
335
00:15:41,860 --> 00:15:43,220
notice what's happening.
336
00:15:43,220 --> 00:15:47,280
Each time that I passed 3/2, I
pass it by less than before
337
00:15:47,280 --> 00:15:49,500
because the terms are getting
smaller in magnitude.
338
00:15:49,500 --> 00:15:54,080
What's happening is and I hope
this crazy little diagram here
339
00:15:54,080 --> 00:15:55,110
serves the purpose.
340
00:15:55,110 --> 00:15:59,910
You see what's happening is
I'm zeroing in on 3/2.
341
00:15:59,910 --> 00:16:03,690
In other words, this particular
rearrangement will
342
00:16:03,690 --> 00:16:07,230
guarantee me that those terms
will add up to 3/2.
343
00:16:07,230 --> 00:16:11,050
Now again, as I said before, 3/2
was not important, though
344
00:16:11,050 --> 00:16:12,430
the arithmetic gets messier.
345
00:16:12,430 --> 00:16:14,960
And again, that's the best
word I can think of.
346
00:16:14,960 --> 00:16:17,150
In other words, I went through
several sheets of paper just
347
00:16:17,150 --> 00:16:19,650
trying to get to the next
stage over here before I
348
00:16:19,650 --> 00:16:21,250
realized it wasn't worth it.
349
00:16:21,250 --> 00:16:23,280
I mean it's something we
can all do on our own.
350
00:16:23,280 --> 00:16:26,490
But the larger number that you
choose, the more terms you're
351
00:16:26,490 --> 00:16:29,420
going to have to add up before
you exceed this.
352
00:16:29,420 --> 00:16:31,110
Don't confuse two things here.
353
00:16:31,110 --> 00:16:34,900
I obviously have to add up an
awful lot of terms of the form
354
00:16:34,900 --> 00:16:40,680
1, 1/3, 1/5, 1/7, 1/9, 1/11
to get, say a million.
355
00:16:40,680 --> 00:16:43,610
But the point is that since
that series diverges,
356
00:16:43,610 --> 00:16:45,820
eventually by going
out far enough--
357
00:16:45,820 --> 00:16:48,180
now far enough might be
billions of terms.
358
00:16:48,180 --> 00:16:52,120
But still a finite number, the
sum will exceed 1 million.
359
00:16:52,120 --> 00:16:53,710
That's the key point.
360
00:16:53,710 --> 00:16:56,040
In other words, I can keep
oscillating around any sum
361
00:16:56,040 --> 00:16:59,040
that I want just by exceeding
it, coming back with a
362
00:16:59,040 --> 00:17:01,500
negative terms, getting less
than that, and alternating
363
00:17:01,500 --> 00:17:02,660
back and forth.
364
00:17:02,660 --> 00:17:06,329
Again, this will be left for
much greater detail for the
365
00:17:06,329 --> 00:17:09,329
supplementary notes
and the exercises.
366
00:17:09,329 --> 00:17:11,730
I'll mention a little bit more
about that in a few minutes.
367
00:17:11,730 --> 00:17:14,680
But the summary so
far is this.
368
00:17:14,680 --> 00:17:17,970
If the series summation 'n' goes
from 1 to infinity, 'a
369
00:17:17,970 --> 00:17:22,200
sub n' is conditionally
convergent, its limit exists.
370
00:17:22,200 --> 00:17:23,869
Let's not forget that.
371
00:17:23,869 --> 00:17:25,260
Its limit exists.
372
00:17:25,260 --> 00:17:28,850
But that limit depends on not
changing the order in which
373
00:17:28,850 --> 00:17:29,820
the terms were given.
374
00:17:29,820 --> 00:17:32,940
In other words, the limit
changes as the order of the
375
00:17:32,940 --> 00:17:34,690
terms is changed.
376
00:17:34,690 --> 00:17:39,420
That is, rearranging the terms
actually changes the series.
377
00:17:39,420 --> 00:17:42,370
And there is nothing in finite
arithmetic that is
378
00:17:42,370 --> 00:17:43,220
comparable to this.
379
00:17:43,220 --> 00:17:46,260
In other words, if you have 50
numbers to add up, or 50
380
00:17:46,260 --> 00:17:49,150
million numbers, or 50 billion
numbers, no matter how you
381
00:17:49,150 --> 00:17:54,880
rearrange those numbers, the
sum exists and is the same
382
00:17:54,880 --> 00:17:58,660
independently of what the
rearrangement is.
383
00:17:58,660 --> 00:18:01,500
This is not comparable
to finite arithmetic.
384
00:18:01,500 --> 00:18:03,940
And the moral is-- and again, I
say this in slang expression
385
00:18:03,940 --> 00:18:05,170
because I want this
to rub off.
386
00:18:05,170 --> 00:18:06,620
I want you to remember this.
387
00:18:06,620 --> 00:18:09,620
Don't monkey with conditional
convergence.
388
00:18:09,620 --> 00:18:13,790
If the series is conditionally
convergent, make sure that you
389
00:18:13,790 --> 00:18:17,366
add the terms in the order
in which they appear.
390
00:18:17,366 --> 00:18:20,360
That if you change the
order, you will
391
00:18:20,360 --> 00:18:21,970
get a different limit.
392
00:18:21,970 --> 00:18:24,660
And what will happen is you'll
get a limit that is the right
393
00:18:24,660 --> 00:18:26,990
answer to the wrong problem.
394
00:18:26,990 --> 00:18:28,450
In other words, changing
the order of the
395
00:18:28,450 --> 00:18:30,870
terms changes the limit.
396
00:18:30,870 --> 00:18:33,370
And this is why conditional
convergence is
397
00:18:33,370 --> 00:18:34,750
particularly annoying.
398
00:18:34,750 --> 00:18:36,990
It means that all of these
things that come natural an
399
00:18:36,990 --> 00:18:39,970
ordinary arithmetic are
lacking in conditional
400
00:18:39,970 --> 00:18:41,540
convergence.
401
00:18:41,540 --> 00:18:44,160
Now, what does the
sequel to this?
402
00:18:44,160 --> 00:18:48,110
The sequel is that all is well
when you have absolute
403
00:18:48,110 --> 00:18:49,840
convergence.
404
00:18:49,840 --> 00:18:51,680
I just wrote this out to make
sure that we have this in
405
00:18:51,680 --> 00:18:52,520
front of us.
406
00:18:52,520 --> 00:18:56,400
The beauty of absolute
convergence is that the sum of
407
00:18:56,400 --> 00:18:59,890
an absolutely convergent series
is the same for every
408
00:18:59,890 --> 00:19:01,770
rearrangement of the terms.
409
00:19:01,770 --> 00:19:05,060
The details are left to the
supplementary notes.
410
00:19:05,060 --> 00:19:09,110
Now, what am I trying to bring
out by all of this?
411
00:19:09,110 --> 00:19:14,270
You see, the beauty of positive
series is that every
412
00:19:14,270 --> 00:19:17,750
time we talk about absolute
convergence, the test involves
413
00:19:17,750 --> 00:19:18,650
a positive series.
414
00:19:18,650 --> 00:19:21,540
In other words, by knowing how
to test positive series for
415
00:19:21,540 --> 00:19:24,830
convergence, we can test
any series for absolute
416
00:19:24,830 --> 00:19:25,800
convergence.
417
00:19:25,800 --> 00:19:28,270
What is the beauty of absolute
convergence?
418
00:19:28,270 --> 00:19:31,790
The beauty of absolute
convergence is that we can
419
00:19:31,790 --> 00:19:34,260
rearrange the terms in any order
that we want if it's
420
00:19:34,260 --> 00:19:36,570
convenient to pick a different
order than another.
421
00:19:36,570 --> 00:19:40,030
And the sum will not depend
on this rearrangement.
422
00:19:40,030 --> 00:19:42,330
You see the point is we are
not saying keep away from
423
00:19:42,330 --> 00:19:44,160
conditionally convergent
series.
424
00:19:44,160 --> 00:19:47,930
In many important applications
you have to come to grips with
425
00:19:47,930 --> 00:19:49,330
conditional convergence.
426
00:19:49,330 --> 00:19:52,270
All we are saying is that if
you want to be able to fool
427
00:19:52,270 --> 00:19:55,620
around numerically with these
series, if you don't have
428
00:19:55,620 --> 00:19:58,590
absolute convergence, you're
in a bit of trouble.
429
00:19:58,590 --> 00:20:01,500
Now you see, the point is that
our textbook does a very good
430
00:20:01,500 --> 00:20:05,030
job in talking about absolute
convergence versus conditional
431
00:20:05,030 --> 00:20:05,840
convergence.
432
00:20:05,840 --> 00:20:09,180
But for some reason, does not
mention the problem of
433
00:20:09,180 --> 00:20:10,900
rearranging terms.
434
00:20:10,900 --> 00:20:13,640
And therefore, much of what I've
talked about today, the
435
00:20:13,640 --> 00:20:16,920
importance of absolute
convergence, is in terms of
436
00:20:16,920 --> 00:20:17,990
rearrangements.
437
00:20:17,990 --> 00:20:20,730
And because this material is
not in the textbook, what I
438
00:20:20,730 --> 00:20:24,900
have elected to do is to put
all of this material that
439
00:20:24,900 --> 00:20:28,420
we've talked about today almost
verbatim except in a
440
00:20:28,420 --> 00:20:31,680
more generalized form, into
the supplementary notes,
441
00:20:31,680 --> 00:20:34,510
supplying whatever proofs are
necessary and whatever
442
00:20:34,510 --> 00:20:36,460
intuitive ideas are necessary.
443
00:20:36,460 --> 00:20:39,350
At any rate, read the
supplementary notes, do the
444
00:20:39,350 --> 00:20:43,370
exercises, and we'll continue
our discussion next time.
445
00:20:43,370 --> 00:20:44,700
And until next time, goodbye.
446
00:20:44,700 --> 00:20:47,780
447
00:20:47,780 --> 00:20:50,310
ANNOUNCER: Funding for the
publication of this video was
448
00:20:50,310 --> 00:20:55,030
provided by the Gabriella and
Paul Rosenbaum Foundation.
449
00:20:55,030 --> 00:20:59,200
Help OCW continue to provide
free and open access to MIT
450
00:20:59,200 --> 00:21:03,400
courses by making a donation
at ocw.mit.edu/donate.
451
00:21:03,400 --> 00:21:08,138