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PROFESSOR: Hi.
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Today we begin our final block
of material in this particular
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00:00:30,770 --> 00:00:34,650
course, and it's the segment
entitled Infinite Series.
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00:00:34,650 --> 00:00:37,510
And perhaps the best way to
motivate this rather difficult
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00:00:37,510 --> 00:00:40,260
block of material is in
terms of the concept
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00:00:40,260 --> 00:00:42,080
of many versus infinite.
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00:00:42,080 --> 00:00:45,190
In many respects, this
particular block could've been
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00:00:45,190 --> 00:00:46,860
given much earlier
in the course.
18
00:00:46,860 --> 00:00:50,140
But somehow or other, until we
have some sort of a feeling as
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00:00:50,140 --> 00:00:53,760
to what infinity really means,
we have a maturity problem in
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00:00:53,760 --> 00:00:55,910
trying to really grasp
the significance of
21
00:00:55,910 --> 00:00:56,850
what's going on.
22
00:00:56,850 --> 00:00:59,120
In fact, in a manner of
speaking, with all of this
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00:00:59,120 --> 00:01:02,130
experience, there may be a
maturity problem in trying to
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00:01:02,130 --> 00:01:03,770
grasp the fundamental ideas.
25
00:01:03,770 --> 00:01:07,000
What I shall do throughout the
material on this block is to
26
00:01:07,000 --> 00:01:10,180
utilize the lectures again to
make sure that the concepts
27
00:01:10,180 --> 00:01:13,720
become crystallized and use the
learning exercises plus
28
00:01:13,720 --> 00:01:17,140
the text plus supplements notes
to make sure that the
29
00:01:17,140 --> 00:01:20,750
details are taken care of
in adequate fashion.
30
00:01:20,750 --> 00:01:23,040
At any rate, I've entitle
today's lecture
31
00:01:23,040 --> 00:01:24,450
'Many Versus Infinite'.
32
00:01:24,450 --> 00:01:27,290
And I thought the best way to
get started on this was to
33
00:01:27,290 --> 00:01:29,990
think of a number that's very
easy to write in terms of
34
00:01:29,990 --> 00:01:31,790
exponential notation.
35
00:01:31,790 --> 00:01:35,560
Let capital 'N' be 10 to the
10 to the 10th power.
36
00:01:35,560 --> 00:01:37,910
10 to the 10, by the way,
is 10 billion, a 1
37
00:01:37,910 --> 00:01:39,270
followed by 10 zeroes.
38
00:01:39,270 --> 00:01:42,030
That's 10 to the 10-billionth
power.
39
00:01:42,030 --> 00:01:44,530
That, of course, means, if
written in place value, that
40
00:01:44,530 --> 00:01:48,200
would be a 1 followed by
10 billion zeroes.
41
00:01:48,200 --> 00:01:50,190
And for those of you who would
like an exercise in
42
00:01:50,190 --> 00:01:52,810
multiplication and long division
and you want to
43
00:01:52,810 --> 00:01:55,790
compute the number of seconds
in a year and what have you,
44
00:01:55,790 --> 00:01:57,970
it turns out without
too much difficulty
45
00:01:57,970 --> 00:01:59,360
that it can be shown.
46
00:01:59,360 --> 00:02:02,990
That to write 1 billion zeroes
at the rate of one per second
47
00:02:02,990 --> 00:02:06,380
would take in the order of
magnitude of some 32 years.
48
00:02:06,380 --> 00:02:10,060
In other words, this number
capital 'N', roughly speaking,
49
00:02:10,060 --> 00:02:12,960
writing it in place value
notation at the rate of one
50
00:02:12,960 --> 00:02:16,640
digit per second would take
320 years to write.
51
00:02:16,640 --> 00:02:18,070
And you say so what?
52
00:02:18,070 --> 00:02:20,810
And the answer is, well, after
you've got out that far-- and
53
00:02:20,810 --> 00:02:22,560
by the way, this is crucial.
54
00:02:22,560 --> 00:02:25,380
320 years is a long time.
55
00:02:25,380 --> 00:02:26,230
I was going to say
it's a lifetime.
56
00:02:26,230 --> 00:02:27,610
It's more than the lifetime.
57
00:02:27,610 --> 00:02:29,860
It's a long time,
but it's finite.
58
00:02:29,860 --> 00:02:32,190
Eventually, the job could
be completed.
59
00:02:32,190 --> 00:02:34,710
But the interesting point is
that once it's completed, the
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00:02:34,710 --> 00:02:38,400
next number in our system is
capital 'N plus 1', capital 'N
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00:02:38,400 --> 00:02:42,920
plus 2', capital 'N plus 3',
where in a sense then, with
62
00:02:42,920 --> 00:02:46,140
'N' as a new reference point,
we're back to the beginning of
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00:02:46,140 --> 00:02:47,030
our number system.
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00:02:47,030 --> 00:02:50,740
In other words, granted that 'N'
is a fantastically large
65
00:02:50,740 --> 00:02:54,020
number, if you wanted to become
wealthy, to own 'N'
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00:02:54,020 --> 00:02:56,810
dollars would more than
realize your dream.
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00:02:56,810 --> 00:03:00,850
But if your aim was to own
infinitely much money, 'N'
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00:03:00,850 --> 00:03:04,220
would be no closer than having
no money at all. 'N' is no
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00:03:04,220 --> 00:03:06,520
nearer the end of a number
system than is
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00:03:06,520 --> 00:03:07,990
the number 1 itself.
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00:03:07,990 --> 00:03:11,020
There is the story that
signifies the difference
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00:03:11,020 --> 00:03:12,670
between many and infinite.
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00:03:12,670 --> 00:03:14,590
And to hammer this point
home, let me give
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00:03:14,590 --> 00:03:16,600
you a few more examples.
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00:03:16,600 --> 00:03:19,740
I cleverly call this additional
examples.
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00:03:19,740 --> 00:03:22,720
We all know that there are just
as many odd numbers and
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00:03:22,720 --> 00:03:23,640
even numbers, right?
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00:03:23,640 --> 00:03:25,600
The odds and the
evens match up.
79
00:03:25,600 --> 00:03:27,630
Now, watch the following
little gimmick.
80
00:03:27,630 --> 00:03:30,450
Write the first two odd numbers,
then the first even
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00:03:30,450 --> 00:03:33,490
number, the next two odd
numbers, then the next even
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00:03:33,490 --> 00:03:35,570
number, the next two
odd numbers, than
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00:03:35,570 --> 00:03:36,780
the next even number.
84
00:03:36,780 --> 00:03:39,080
And go on like this as
long as you want.
85
00:03:39,080 --> 00:03:43,510
And no matter where we stop,
even if we go to the 10 to the
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00:03:43,510 --> 00:03:47,390
10 to the 10th term, no matter
what even number we stop at,
87
00:03:47,390 --> 00:03:50,470
there will always be twice as
many odd numbers written on
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00:03:50,470 --> 00:03:53,450
the board as there would
be even numbers.
89
00:03:53,450 --> 00:03:57,090
In other words, even though in
the long run in terms of the
90
00:03:57,090 --> 00:04:00,070
infinity of each there are as
many odds and evens, if we
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00:04:00,070 --> 00:04:03,650
stop this process at any finite
time no matter how far
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00:04:03,650 --> 00:04:06,130
out, there will always
be twice as many odds
93
00:04:06,130 --> 00:04:07,130
as there are evens.
94
00:04:07,130 --> 00:04:10,290
In fact, if you want to compound
this little dilemma,
95
00:04:10,290 --> 00:04:12,680
write the first two evens, then
an odd, in other words,
96
00:04:12,680 --> 00:04:18,390
2, 4, 1, 6, 8, 3, 10, 12, 5, and
you can get twice as many
97
00:04:18,390 --> 00:04:20,490
evens as there are
odds, et cetera.
98
00:04:20,490 --> 00:04:22,890
And the whole argument
again hinges on what?
99
00:04:22,890 --> 00:04:26,560
Confusing the concept of
going out very far
100
00:04:26,560 --> 00:04:28,940
with going out endlessly.
101
00:04:28,940 --> 00:04:30,480
Oh, let me give you another
example or two.
102
00:04:30,480 --> 00:04:32,940
I just want to throw these
around so you at least get the
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00:04:32,940 --> 00:04:36,850
mood created as to what we're
really dealing with right now.
104
00:04:36,850 --> 00:04:40,460
Let's take the endless sequence
of numbers, the sum,
105
00:04:40,460 --> 00:04:44,370
1 plus 'minus 1' plus 1
plus 'minus 1', and
106
00:04:44,370 --> 00:04:45,900
say let's go on forever.
107
00:04:45,900 --> 00:04:48,050
What will this sum be?
108
00:04:48,050 --> 00:04:49,760
Now, lookit, one
way of grouping
109
00:04:49,760 --> 00:04:51,750
these terms is in twos.
110
00:04:51,750 --> 00:04:54,550
In other words, we'll start with
the first two terms, the
111
00:04:54,550 --> 00:04:55,330
next two terms.
112
00:04:55,330 --> 00:04:59,460
In other words, we can write
this as 1 plus minus 1 plus 1
113
00:04:59,460 --> 00:05:00,700
plus minus 1.
114
00:05:00,700 --> 00:05:04,200
And writing it this way, we can
see that each term adds up
115
00:05:04,200 --> 00:05:07,210
to 0, and the infinite
sum would be 0.
116
00:05:07,210 --> 00:05:10,430
On the other hand, if we now
leave the first term alone and
117
00:05:10,430 --> 00:05:15,030
now start grouping the remaining
terms in twos, we
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00:05:15,030 --> 00:05:17,440
find that the infinite
sum is 1.
119
00:05:17,440 --> 00:05:20,200
Now, we're not going to argue
that something is fishy here.
120
00:05:20,200 --> 00:05:21,340
We're not going to say
I wonder which
121
00:05:21,340 --> 00:05:22,170
is the right answer.
122
00:05:22,170 --> 00:05:26,790
What we have shown without fear
of contradiction is that
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00:05:26,790 --> 00:05:29,600
the answer that you get when you
add infinitely many terms
124
00:05:29,600 --> 00:05:33,060
does depend on how you group
them, unlike the situation of
125
00:05:33,060 --> 00:05:35,560
what happens when you add
finitely many terms.
126
00:05:35,560 --> 00:05:38,730
In other words, notice the need
for order as well as the
127
00:05:38,730 --> 00:05:41,530
terms themselves when
you have a sum of
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00:05:41,530 --> 00:05:42,960
infinitely many terms.
129
00:05:42,960 --> 00:05:47,000
And the key point is don't be
upset when you find out that
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00:05:47,000 --> 00:05:48,480
your intuition is defied here.
131
00:05:48,480 --> 00:05:50,910
You say this doesn't
seem real to me.
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00:05:50,910 --> 00:05:52,880
It seems intuitively false.
133
00:05:52,880 --> 00:05:56,110
The point is our intuition
is defied.
134
00:05:56,110 --> 00:05:56,850
Why?
135
00:05:56,850 --> 00:05:58,710
Because it doesn't apply.
136
00:05:58,710 --> 00:06:00,310
And why doesn't it apply?
137
00:06:00,310 --> 00:06:03,810
It doesn't apply because our
intuition is based on
138
00:06:03,810 --> 00:06:08,190
visualizing large but finite
amounts, not based on
139
00:06:08,190 --> 00:06:10,380
visualizing infinity.
140
00:06:10,380 --> 00:06:13,250
You see, all of these paradoxes
stem, because in our
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00:06:13,250 --> 00:06:16,510
mind, we're trying to visualize
infinity as meaning
142
00:06:16,510 --> 00:06:18,010
the same as very large.
143
00:06:18,010 --> 00:06:20,470
Well, you know, now we come to
a very important crossroad.
144
00:06:20,470 --> 00:06:23,450
After all, if infinity is going
to be this difficult a
145
00:06:23,450 --> 00:06:26,770
concept to handle, let's get
rid of it the easy way.
146
00:06:26,770 --> 00:06:28,720
Let's refuse to study it.
147
00:06:28,720 --> 00:06:30,560
That's one way of solving
problems.
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00:06:30,560 --> 00:06:33,140
It's what I call the right wing
conservative educational
149
00:06:33,140 --> 00:06:33,700
philosophy.
150
00:06:33,700 --> 00:06:36,060
If you don't like something,
throw it out.
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00:06:36,060 --> 00:06:37,870
The only trouble
is we need it.
152
00:06:37,870 --> 00:06:40,160
For example, why
do we need it?
153
00:06:40,160 --> 00:06:41,380
See, why deal with
infinite sums?
154
00:06:41,380 --> 00:06:42,430
Well, because we need them.
155
00:06:42,430 --> 00:06:44,700
Among other places, we've
already used them.
156
00:06:44,700 --> 00:06:46,430
For example, in computing
areas.
157
00:06:46,430 --> 00:06:48,780
We've taken a limit as
'N' goes to infinity.
158
00:06:48,780 --> 00:06:53,240
Summation, 'k' goes from 1 to
'n', 'f of 'c sub k'' 'delta
159
00:06:53,240 --> 00:06:54,630
x', you see.
160
00:06:54,630 --> 00:06:55,840
And we need this limit.
161
00:06:55,840 --> 00:06:58,820
And so the question comes up,
how shall we add infinitely
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00:06:58,820 --> 00:06:59,390
many terms?
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00:06:59,390 --> 00:07:00,560
We have a choice now.
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00:07:00,560 --> 00:07:02,480
We can throw the thing
out, but we don't
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00:07:02,480 --> 00:07:03,180
want to throw it out.
166
00:07:03,180 --> 00:07:04,280
We need it.
167
00:07:04,280 --> 00:07:06,860
So the question is how shall we
add infinitely many terms?
168
00:07:06,860 --> 00:07:09,690
And even though we know that
our intuition can get us in
169
00:07:09,690 --> 00:07:13,260
trouble, we do have nothing
else to begin with.
170
00:07:13,260 --> 00:07:16,880
So we say OK, let's mimic what
happened in the finite case
171
00:07:16,880 --> 00:07:20,380
and see if we can't extend that
in a plausible way to
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00:07:20,380 --> 00:07:21,720
cover the infinite case.
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00:07:21,720 --> 00:07:24,900
Let me pick a particularly
straightforward example.
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00:07:24,900 --> 00:07:27,840
Let's suppose I have the three
numbers which I'll call 'a sub
175
00:07:27,840 --> 00:07:31,730
1', 'a sub 2', and 'a sub 3',
where 'a sub 1' will be 1/2,
176
00:07:31,730 --> 00:07:35,310
'a sub 2' will be 1/4, and
'a sub 3' will be 1/8.
177
00:07:35,310 --> 00:07:37,640
In other words, just for reasons
of identification
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00:07:37,640 --> 00:07:41,270
later on in what I'm going to be
doing, each term is half of
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00:07:41,270 --> 00:07:43,190
the previous one.
180
00:07:43,190 --> 00:07:45,680
Now, I want to find the sum
of these three terms.
181
00:07:45,680 --> 00:07:49,320
I want to find 'a1' plus
'a2' plus 'a3'.
182
00:07:49,320 --> 00:07:51,750
Now, colloquially, we just say,
oh, that's 1/2 plus 1/4
183
00:07:51,750 --> 00:07:54,270
plus 1/8, and I'll
just add them up.
184
00:07:54,270 --> 00:07:55,650
But do you remember how
you learned to add?
185
00:07:55,650 --> 00:07:57,470
You may not have paid attention
to it, but you
186
00:07:57,470 --> 00:07:58,780
learned to add as a sequence.
187
00:07:58,780 --> 00:08:00,750
You said I'll add
the first one.
188
00:08:00,750 --> 00:08:04,030
Then the first plus the second
gives me a number.
189
00:08:04,030 --> 00:08:05,850
That's my second partial sum.
190
00:08:05,850 --> 00:08:07,500
Then I'll add on the
third number.
191
00:08:07,500 --> 00:08:09,830
That will give me my
third partial sum.
192
00:08:09,830 --> 00:08:11,940
Then I have no more
numbers to add.
193
00:08:11,940 --> 00:08:16,460
Consequently, my third partial
sum is by definition the sum
194
00:08:16,460 --> 00:08:17,840
of these three numbers.
195
00:08:17,840 --> 00:08:21,160
Writing it more symbolically,
we say lookit, the first
196
00:08:21,160 --> 00:08:23,720
partial sum, 's sub
1', is 1/2.
197
00:08:23,720 --> 00:08:26,430
The second partial sum
as 1/2 plus 1/4.
198
00:08:26,430 --> 00:08:27,990
Another way of saying
that is what?
199
00:08:27,990 --> 00:08:30,550
It's the first partial
sums plus the next
200
00:08:30,550 --> 00:08:31,860
term, which is 1/4.
201
00:08:31,860 --> 00:08:34,549
1/2 plus 1/4 is 3/4.
202
00:08:34,549 --> 00:08:38,049
Then we said OK, the third
partial sum is what we had
203
00:08:38,049 --> 00:08:42,820
before, namely, 3/4, plus the
next term, which was 1/8, and
204
00:08:42,820 --> 00:08:45,070
that gives rise to 7/8.
205
00:08:45,070 --> 00:08:49,320
In other words, we said let's
form 'a1', then 'a1 plus a2',
206
00:08:49,320 --> 00:08:51,640
'a1 plus a2 plus a3'.
207
00:08:51,640 --> 00:08:54,420
And when we finally finished
with our sequence of partial
208
00:08:54,420 --> 00:08:57,760
sums, the last partial
sum was the answer.
209
00:08:57,760 --> 00:09:01,330
And by the way, let me take time
out here to hit home at
210
00:09:01,330 --> 00:09:04,220
the most important point, the
point that I think is
211
00:09:04,220 --> 00:09:06,460
extremely crucial as a starting
point if we're going
212
00:09:06,460 --> 00:09:09,200
to understand what this whole
block is all about.
213
00:09:09,200 --> 00:09:12,135
It's to distinguish between
a series and a sequence.
214
00:09:12,135 --> 00:09:14,370
And I'll have much more to
say about this in the
215
00:09:14,370 --> 00:09:15,430
supplementary notes.
216
00:09:15,430 --> 00:09:17,500
But for now, think
of it this way.
217
00:09:17,500 --> 00:09:20,530
A series is a sum of terms.
218
00:09:20,530 --> 00:09:22,960
A sequence is just a
listing of terms.
219
00:09:22,960 --> 00:09:25,800
In other words, in this
particular problem, do not
220
00:09:25,800 --> 00:09:29,960
confuse the role of the 'a's
with the role of the 's's.
221
00:09:29,960 --> 00:09:33,240
Notice that the a's refer
to the sequence of
222
00:09:33,240 --> 00:09:34,850
numbers being added.
223
00:09:34,850 --> 00:09:36,600
In other words, the
'a's were what?
224
00:09:36,600 --> 00:09:39,170
They were 1/2, 1/4 and 1/8.
225
00:09:39,170 --> 00:09:41,440
These were the three numbers
being added.
226
00:09:41,440 --> 00:09:44,660
Notice that the 's's were
the partial sums.
227
00:09:44,660 --> 00:09:48,690
In other words, the partial sums
form the sequence 's1',
228
00:09:48,690 --> 00:09:50,430
's2', 's3'.
229
00:09:50,430 --> 00:09:52,780
And to refresh your memories
on this, that would be the
230
00:09:52,780 --> 00:09:53,770
sequence what?
231
00:09:53,770 --> 00:09:57,150
1/2, 3/4, 7/8.
232
00:09:57,150 --> 00:09:59,260
In other words, this was the sum
of the first number that
233
00:09:59,260 --> 00:10:00,060
you were adding.
234
00:10:00,060 --> 00:10:05,340
3/4 was the sum of first two,
and 7/8 was the sum of all
235
00:10:05,340 --> 00:10:06,240
three of them.
236
00:10:06,240 --> 00:10:10,330
And notice, by the way, the last
partial sum, 's sub 3',
237
00:10:10,330 --> 00:10:13,360
the sum was defined to
be the last partial
238
00:10:13,360 --> 00:10:14,910
sum, and that is what?
239
00:10:14,910 --> 00:10:15,790
The number--
240
00:10:15,790 --> 00:10:17,140
this is very, very crucial.
241
00:10:17,140 --> 00:10:21,500
1/2 plus 1/4 plus 1/8 is the sum
of three numbers, but it's
242
00:10:21,500 --> 00:10:24,480
one number, and that number
is called 7/8.
243
00:10:24,480 --> 00:10:26,280
OK, you see what we're
talking about now?
244
00:10:26,280 --> 00:10:27,960
We're looking at a
bunch of terms.
245
00:10:27,960 --> 00:10:30,390
We're adding them up, and
we see how the sum
246
00:10:30,390 --> 00:10:32,110
changes with each term.
247
00:10:32,110 --> 00:10:35,150
In fact, in terms of a very
trivial analogy, think of an
248
00:10:35,150 --> 00:10:37,450
adding machine.
249
00:10:37,450 --> 00:10:41,050
As you punch numbers in, the
's's are the sums that you see
250
00:10:41,050 --> 00:10:44,730
being read as your total sum,
whereas the a's are the
251
00:10:44,730 --> 00:10:48,450
individual numbers being punched
in to add up, OK?
252
00:10:48,450 --> 00:10:49,800
I hope that's a trivial
example.
253
00:10:49,800 --> 00:10:52,840
As I listen to myself saying it,
it sounds like I made it
254
00:10:52,840 --> 00:10:54,230
harder than it really is.
255
00:10:54,230 --> 00:10:57,230
At any rate, let's generalize
this particular problem.
256
00:10:57,230 --> 00:11:00,190
Let's suppose now instead of
wanting to add 1/2 plus 1/4
257
00:11:00,190 --> 00:11:04,320
plus 1/8, we want to add up
the first 'n' terms of the
258
00:11:04,320 --> 00:11:06,710
form 1/2 plus 1/4 plus 1/8.
259
00:11:06,710 --> 00:11:09,060
In other words, let the n-th
term that we're going to add,
260
00:11:09,060 --> 00:11:12,190
'a sub n', be '1 over
'2 to the n''.
261
00:11:12,190 --> 00:11:16,280
Then the sum, the n-th partial
sum here, the sum of these 'n'
262
00:11:16,280 --> 00:11:18,570
terms is, of course,
what? 'a1' plus, et
263
00:11:18,570 --> 00:11:19,880
cetera, 'a sub n'.
264
00:11:19,880 --> 00:11:23,950
That turns out to be 1/2 plus
1/4 plus, et cetera, '1 over
265
00:11:23,950 --> 00:11:25,680
'2 to the n''.
266
00:11:25,680 --> 00:11:29,670
And by the way, rather than
take time to develop this
267
00:11:29,670 --> 00:11:32,780
recipe over here, I thought you
might like to see another
268
00:11:32,780 --> 00:11:35,690
place that might be interesting
to review
269
00:11:35,690 --> 00:11:37,260
mathematical induction.
270
00:11:37,260 --> 00:11:40,280
If you'll bear with me and just
come back over here where
271
00:11:40,280 --> 00:11:44,660
we were computing these partial
sums, notice that in
272
00:11:44,660 --> 00:11:48,130
each of these partial sums,
notice that your denominator
273
00:11:48,130 --> 00:11:52,160
is always 2 raised to the same
power as this subscript.
274
00:11:52,160 --> 00:11:54,390
See, 2 the first power is 2.
275
00:11:54,390 --> 00:11:57,030
2 to the second power is 4.
276
00:11:57,030 --> 00:11:59,380
2 to the third power is 8.
277
00:11:59,380 --> 00:12:01,940
In other words, if your
subscript is n, your
278
00:12:01,940 --> 00:12:04,240
denominator is '2 to the n'.
279
00:12:04,240 --> 00:12:07,190
Notice that your numerator is
always one less than your
280
00:12:07,190 --> 00:12:08,080
denominator.
281
00:12:08,080 --> 00:12:11,180
In other words, if your
denominator is '2 to the n',
282
00:12:11,180 --> 00:12:14,060
the numerator is '2 to
the 'n minus 1''.
283
00:12:14,060 --> 00:12:17,570
And once we suspect this, this
particular result can be
284
00:12:17,570 --> 00:12:19,040
proven by induction.
285
00:12:19,040 --> 00:12:20,780
I won't take the time
to do this here.
286
00:12:20,780 --> 00:12:23,690
What I will take the time to
do is to observe that this
287
00:12:23,690 --> 00:12:27,250
particular sum can be written
more conveniently if we divide
288
00:12:27,250 --> 00:12:30,260
through by '2 to the n-th'
to get 1 minus
289
00:12:30,260 --> 00:12:31,990
'1 over '2 the n''.
290
00:12:31,990 --> 00:12:35,440
For example, suppose we wanted
to add up the 10 numbers.
291
00:12:35,440 --> 00:12:36,570
I say 10 numbers here.
292
00:12:36,570 --> 00:12:39,050
2 to the 10th is 1.024.
293
00:12:39,050 --> 00:12:43,190
But according to this recipe,
1/2 plus 1/4 plus 1/8 plus
294
00:12:43,190 --> 00:12:48,405
1/16 plus, et cetera, et cetera,
plus 1/1,024 would add
295
00:12:48,405 --> 00:12:49,390
up to be what?
296
00:12:49,390 --> 00:12:53,820
1 minus 1/1,024.
297
00:12:53,820 --> 00:12:59,300
In other words, this would be
1,023/1,024, which seems to be
298
00:12:59,300 --> 00:13:00,700
pretty close to 1.
299
00:13:00,700 --> 00:13:03,390
In fact, you can begin to
suspect that as 'n' gets
300
00:13:03,390 --> 00:13:06,660
arbitrarily large, 's sub
n' gets arbitrarily
301
00:13:06,660 --> 00:13:08,100
close to 1 in value.
302
00:13:08,100 --> 00:13:09,750
I'm just talking fairly
intuitively
303
00:13:09,750 --> 00:13:11,550
for the time being.
304
00:13:11,550 --> 00:13:14,460
But, you see, the major question
now is suppose you
305
00:13:14,460 --> 00:13:16,410
elect not to stop at that.
306
00:13:16,410 --> 00:13:17,860
And you see, this is
a very key point.
307
00:13:17,860 --> 00:13:21,810
We've already seen how the whole
world seems to change as
308
00:13:21,810 --> 00:13:25,720
soon as you say let's never
stop as opposed to saying
309
00:13:25,720 --> 00:13:27,650
let's go out as far
as you want.
310
00:13:27,650 --> 00:13:29,810
See, if we now say what
happens if you go on
311
00:13:29,810 --> 00:13:31,010
endlessly over here?
312
00:13:31,010 --> 00:13:34,190
Well, it becomes very natural
to say lookit, the n-th
313
00:13:34,190 --> 00:13:37,920
partial sum was 1 minus '1
over '2 to the n-th'.
314
00:13:37,920 --> 00:13:40,360
In the case where you were
adding up a finite number of
315
00:13:40,360 --> 00:13:44,040
terms, when you came to the last
partial sum, that was by
316
00:13:44,040 --> 00:13:45,550
definition your answer.
317
00:13:45,550 --> 00:13:47,970
Now, what we're saying is
lookit, because we have
318
00:13:47,970 --> 00:13:51,520
infinitely many terms to add,
there is no last partial sum.
319
00:13:51,520 --> 00:13:54,450
And so what we say is lookit,
instead of the last term,
320
00:13:54,450 --> 00:13:57,210
since there is no last term,
why don't we just take the
321
00:13:57,210 --> 00:14:01,160
limit of the n-th partial sum
as 'n' goes to infinity.
322
00:14:01,160 --> 00:14:03,605
In other words, in this
particular case, notice that
323
00:14:03,605 --> 00:14:07,220
as 'n' approaches infinity, 1
minus '1 over '2 to the n''
324
00:14:07,220 --> 00:14:10,910
approaches 1, and we then
define the infinite sum,
325
00:14:10,910 --> 00:14:12,010
meaning what?
326
00:14:12,010 --> 00:14:14,600
I write it this way: as sigma
'n' goes from 1 to infinity,
327
00:14:14,600 --> 00:14:16,050
'1 over '2 to the n''.
328
00:14:16,050 --> 00:14:17,150
It really means what?
329
00:14:17,150 --> 00:14:20,850
The limit as 'n' goes to
infinity: 1/2 plus 1/4 plus
330
00:14:20,850 --> 00:14:22,410
1/8 plus 1/16--
331
00:14:22,410 --> 00:14:24,190
endlessly--
332
00:14:24,190 --> 00:14:28,380
that that limit is 1, and we
define that to be the sum.
333
00:14:28,380 --> 00:14:31,730
And again, as I say, I'm going
to write that in greatly more
334
00:14:31,730 --> 00:14:34,160
detail in the notes, and
also we'll have many
335
00:14:34,160 --> 00:14:35,360
exercises on this.
336
00:14:35,360 --> 00:14:38,830
I just wanted you to see how we
get to infinite sums, which
337
00:14:38,830 --> 00:14:41,600
are called series by
generalizing what happens in
338
00:14:41,600 --> 00:14:42,930
the finite case.
339
00:14:42,930 --> 00:14:45,350
And because this may seem a
little vague to you, let me
340
00:14:45,350 --> 00:14:47,800
give you a pictorial
representation
341
00:14:47,800 --> 00:14:49,930
of this same thing.
342
00:14:49,930 --> 00:14:52,380
You see, what's happening
here is this.
343
00:14:52,380 --> 00:14:58,240
Draw a little circle around
1 of bandwidth epsilon.
344
00:14:58,240 --> 00:15:00,900
In other words, let's mark off
an interval epsilon on
345
00:15:00,900 --> 00:15:02,080
either side of 1.
346
00:15:02,080 --> 00:15:05,270
And let's call this point
here 1 minus epsilon.
347
00:15:05,270 --> 00:15:07,870
Let's call this point
here 1 plus epsilon.
348
00:15:07,870 --> 00:15:11,140
And what we're saying about
our partial sums is this.
349
00:15:11,140 --> 00:15:13,570
That when you start off and
you're adding up terms here,
350
00:15:13,570 --> 00:15:14,770
you have 1/2.
351
00:15:14,770 --> 00:15:18,720
1/2 plus 1/4 brings
you over to 3/4.
352
00:15:18,720 --> 00:15:21,420
The next possible sum
is 7/8, et cetera.
353
00:15:21,420 --> 00:15:24,870
And all we're saying is that
these terms get arbitrarily
354
00:15:24,870 --> 00:15:28,240
close to 1 in value, meaning
that after a while--
355
00:15:28,240 --> 00:15:30,450
and I'll define more rigorously
what after a while
356
00:15:30,450 --> 00:15:31,490
means in a moment--
357
00:15:31,490 --> 00:15:35,350
all of the 's sub n's are
within epsilon of 1.
358
00:15:35,350 --> 00:15:38,590
After a while, all of your
partial sums are in here.
359
00:15:38,590 --> 00:15:41,330
And what you mean by after a
while certainly depends on how
360
00:15:41,330 --> 00:15:42,260
big epsilon is.
361
00:15:42,260 --> 00:15:45,590
In other words, the smaller
the bandwidth you allow
362
00:15:45,590 --> 00:15:49,350
yourself, the more terms you
may have to take before you
363
00:15:49,350 --> 00:15:50,730
get within the tolerance limits
364
00:15:50,730 --> 00:15:52,270
that you allow yourself.
365
00:15:52,270 --> 00:15:55,500
In any event, going back to
something that we've been
366
00:15:55,500 --> 00:15:57,470
using for a long
time, our basic
367
00:15:57,470 --> 00:15:58,980
definition is the following.
368
00:15:58,980 --> 00:16:01,890
If you have an infinite
sequence, say, a collection of
369
00:16:01,890 --> 00:16:05,077
terms 'b sub n', in other words,
'b1', 'b2', 'b3', et
370
00:16:05,077 --> 00:16:09,900
cetera, without end, we say that
that sequence converges
371
00:16:09,900 --> 00:16:13,390
to the limit 'L' written the
limit of 'b sub n' as 'n'
372
00:16:13,390 --> 00:16:17,360
approaches infinity equals 'L',
if and only if for every
373
00:16:17,360 --> 00:16:21,510
epsilon greater than 0 we can
find a number 'N' which
374
00:16:21,510 --> 00:16:22,910
depends on epsilon--
375
00:16:22,910 --> 00:16:25,480
notice the notation here: 'N'
as a function of epsilon--
376
00:16:25,480 --> 00:16:29,280
such that whenever little 'n'
is greater than capital 'N',
377
00:16:29,280 --> 00:16:31,740
the absolute value of
'a sub n' minus 'L'
378
00:16:31,740 --> 00:16:33,050
is less than epsilon.
379
00:16:33,050 --> 00:16:35,755
And, you see, again, you may
wonder how in the world that
380
00:16:35,755 --> 00:16:36,920
you're going to remember this.
381
00:16:36,920 --> 00:16:39,690
If you memorize this, I
guarantee you, in two day's
382
00:16:39,690 --> 00:16:42,020
time, you'll have to
memorize it again.
383
00:16:42,020 --> 00:16:44,790
I also hope you have enough
faith in me to recognize I
384
00:16:44,790 --> 00:16:46,110
didn't memorize this.
385
00:16:46,110 --> 00:16:48,060
There is a feeling that
one gets for this.
386
00:16:48,060 --> 00:16:50,090
And let me give you what
that feeling is.
387
00:16:50,090 --> 00:16:53,140
Again, in terms of a picture,
what it means-- well, I'll
388
00:16:53,140 --> 00:16:55,720
change these to 'a's now because
that's the symbols
389
00:16:55,720 --> 00:16:58,170
that we've been using
before in terms of
390
00:16:58,170 --> 00:16:59,005
the sequence of terms.
391
00:16:59,005 --> 00:17:01,220
What we really mean-- and I
don't care what symbol you
392
00:17:01,220 --> 00:17:02,270
really use here--
393
00:17:02,270 --> 00:17:06,060
is if you want to talk about the
limit of 'a sub n' as 'n'
394
00:17:06,060 --> 00:17:08,930
approaches infinity, if that
limit equals 'L', what the
395
00:17:08,930 --> 00:17:10,940
rigorous definition
says is this.
396
00:17:10,940 --> 00:17:14,150
Draw yourself an interval
around 'L' of bandwidth
397
00:17:14,150 --> 00:17:15,740
epsilon, in other words,
from 'L minus
398
00:17:15,740 --> 00:17:18,089
epsilon' to 'L plus epsilon'.
399
00:17:18,089 --> 00:17:20,970
And what this thing says is that
beyond a certain term,
400
00:17:20,970 --> 00:17:24,250
say, the capital N-th term,
every term beyond this certain
401
00:17:24,250 --> 00:17:26,700
one is in here.
402
00:17:26,700 --> 00:17:29,790
403
00:17:29,790 --> 00:17:35,240
Well, all 'a n's are in here if
'n' is sufficiently large.
404
00:17:35,240 --> 00:17:37,100
I don't know if you can read
that very well, but just
405
00:17:37,100 --> 00:17:38,020
listen to what I'm saying.
406
00:17:38,020 --> 00:17:42,060
All of the terms are in here if
'n' is sufficiently large.
407
00:17:42,060 --> 00:17:45,820
What this means again is that
to all intents and purposes,
408
00:17:45,820 --> 00:17:49,440
if you think of this bandwidth
as giving you a dot, see, a
409
00:17:49,440 --> 00:17:53,160
thick dot here where the
endpoints are 'L minus
410
00:17:53,160 --> 00:17:56,520
epsilon' and 'L plus epsilon',
what we're saying is lookit,
411
00:17:56,520 --> 00:17:59,600
after a certain term, the way
I've drawn here, after the
412
00:17:59,600 --> 00:18:02,290
fifth term, all the remaining
terms of my
413
00:18:02,290 --> 00:18:04,310
sequence are in here.
414
00:18:04,310 --> 00:18:07,740
By the way, notice the role
of the subscripts here.
415
00:18:07,740 --> 00:18:10,330
All the subscript tells
you is where the term
416
00:18:10,330 --> 00:18:12,550
appears in your sequence.
417
00:18:12,550 --> 00:18:15,300
For example, the third term in
your sequence could be a
418
00:18:15,300 --> 00:18:18,040
smaller number than the second
term of your sequence.
419
00:18:18,040 --> 00:18:21,210
Do not confuse the size of the
terms with the subscripts.
420
00:18:21,210 --> 00:18:24,170
The subscripts order the terms,
but the third term in
421
00:18:24,170 --> 00:18:27,790
the sequence can be less than
in size than the second term
422
00:18:27,790 --> 00:18:28,560
in the sequence.
423
00:18:28,560 --> 00:18:30,170
But again, I'll talk
about that in more
424
00:18:30,170 --> 00:18:31,280
detail in the notes.
425
00:18:31,280 --> 00:18:36,020
The point that I want you to
see is that in concept what
426
00:18:36,020 --> 00:18:37,840
limit does is the following.
427
00:18:37,840 --> 00:18:42,750
Limit is to an infinite sequence
as last term is to a
428
00:18:42,750 --> 00:18:44,200
finite sequence.
429
00:18:44,200 --> 00:18:48,320
In other words, a limit replaces
infinitely many
430
00:18:48,320 --> 00:18:52,720
points by a finite number
of points plus a dot.
431
00:18:52,720 --> 00:18:55,780
You see, going back to this
example here, how many 'a sub
432
00:18:55,780 --> 00:18:56,710
n's were there?
433
00:18:56,710 --> 00:18:58,760
Well, there were infinitely
many.
434
00:18:58,760 --> 00:19:01,530
Well, to keep track of these
infinitely many, what do I
435
00:19:01,530 --> 00:19:02,780
have to keep track of now?
436
00:19:02,780 --> 00:19:07,770
Well, in this diagram, the first
five 'a's plus this dot,
437
00:19:07,770 --> 00:19:12,290
because you see, every one of
my infinitely many 'a's past
438
00:19:12,290 --> 00:19:16,170
the fifth one is inside
this dot, you see?
439
00:19:16,170 --> 00:19:18,450
So in other words then,
what's happened?
440
00:19:18,450 --> 00:19:20,170
The thing that had to happen.
441
00:19:20,170 --> 00:19:22,670
We had to deal with infinite
sequences.
442
00:19:22,670 --> 00:19:26,200
We saw the big philosophic
difference between infinitely
443
00:19:26,200 --> 00:19:28,630
many and just large.
444
00:19:28,630 --> 00:19:33,550
And so our definition of limit
had to be such that we could
445
00:19:33,550 --> 00:19:37,930
reduce in a way that was
compatible with our intuition
446
00:19:37,930 --> 00:19:42,060
the concept of infinitely many
points to a finite number.
447
00:19:42,060 --> 00:19:45,430
Because, you see, as I'll show
you in the notes also, all of
448
00:19:45,430 --> 00:19:49,300
our arithmetic is geared for
just a finite number of
449
00:19:49,300 --> 00:19:50,610
operations.
450
00:19:50,610 --> 00:19:52,280
See, this is why
this definition
451
00:19:52,280 --> 00:19:53,920
of limit is so crucial.
452
00:19:53,920 --> 00:19:57,360
Again, you may notice, and I'll
remind you of this also
453
00:19:57,360 --> 00:20:00,960
in the exercises, that
structurally this definition
454
00:20:00,960 --> 00:20:04,040
of limit is the same as the
limit that we use when we
455
00:20:04,040 --> 00:20:07,000
talked about the limit of 'f of
x', as 'x' approaches 'a',
456
00:20:07,000 --> 00:20:07,880
equals 'L'.
457
00:20:07,880 --> 00:20:12,060
The absolute value signs have
the same properties as before.
458
00:20:12,060 --> 00:20:14,470
And by the way, before I go
on, let me just remind you
459
00:20:14,470 --> 00:20:17,810
again of one more thing while
I'm talking that way.
460
00:20:17,810 --> 00:20:20,950
Instead of memorizing this,
remember how you read this.
461
00:20:20,950 --> 00:20:24,180
The absolute value of 'a sub
n' minus 'L' is less than
462
00:20:24,180 --> 00:20:25,660
epsilon means what?
463
00:20:25,660 --> 00:20:28,820
That 'a sub n' is within
epsilon of 'L'.
464
00:20:28,820 --> 00:20:30,710
That's what we use
in our diagram.
465
00:20:30,710 --> 00:20:32,740
But it seems to me I forgot
to mention this.
466
00:20:32,740 --> 00:20:34,240
And I want you to
see that what?
467
00:20:34,240 --> 00:20:37,790
The key building block
analytically is the absolute
468
00:20:37,790 --> 00:20:40,690
value, and the meaning of
absolute value is the same
469
00:20:40,690 --> 00:20:45,210
here as it was in blocks one
and two of our course.
470
00:20:45,210 --> 00:20:49,010
So what I'm driving at is that
the same limit theorems that
471
00:20:49,010 --> 00:20:52,260
we've been able to use up
until now still apply.
472
00:20:52,260 --> 00:20:54,640
Oh, by means of an example.
473
00:20:54,640 --> 00:20:57,830
Suppose I have the limit as 'n'
approaches infinity, '2n
474
00:20:57,830 --> 00:21:01,110
plus 3' over '5n plus 7'.
475
00:21:01,110 --> 00:21:03,130
Notice that I can divide
numerator and denominator
476
00:21:03,130 --> 00:21:04,190
through by 'n'.
477
00:21:04,190 --> 00:21:07,100
If I do that, I have the limit
as 'n' approaches infinity.
478
00:21:07,100 --> 00:21:11,060
'2 plus '3/n'' over
'5 plus '7/n''.
479
00:21:11,060 --> 00:21:13,910
Now using the fact that the
limit of a sum is the sum of
480
00:21:13,910 --> 00:21:16,380
the limits, the limit of a
quotient is the quotient of
481
00:21:16,380 --> 00:21:21,320
the limits, the limit of '1/n'
as 'n' goes to infinity is 0.
482
00:21:21,320 --> 00:21:24,190
Notice that I can use the limit
theorems to conclude
483
00:21:24,190 --> 00:21:27,450
that the limit of this
particular sequence is 2/5.
484
00:21:27,450 --> 00:21:31,980
If I wanted to, the same ways
we did in block one, block
485
00:21:31,980 --> 00:21:35,710
two, where we're talking about
limits, given an epsilon, I
486
00:21:35,710 --> 00:21:39,360
can actually exhibit how far out
I have to go before each
487
00:21:39,360 --> 00:21:41,800
of the terms in this sequence
is within that
488
00:21:41,800 --> 00:21:44,180
given epsilon of 2/5.
489
00:21:44,180 --> 00:21:48,310
By the way, again to emphasize
once more, because this is so
490
00:21:48,310 --> 00:21:52,230
important, the difference
between an infinite sum and an
491
00:21:52,230 --> 00:21:55,560
infinite sequence, observe that
whereas the limit of the
492
00:21:55,560 --> 00:22:01,070
sequence of terms '2n plus 3'
over '5n plus 7' is 2/5, the
493
00:22:01,070 --> 00:22:07,360
infinite sum composed of the
terms of the form '2n plus 3'
494
00:22:07,360 --> 00:22:13,240
over '5n plus 7' is infinity
since after a while each term
495
00:22:13,240 --> 00:22:16,080
that you're adding here
behaves like 2/5.
496
00:22:16,080 --> 00:22:18,390
In other words, if you write
this thing out to see what
497
00:22:18,390 --> 00:22:20,550
this means, pick 'n' to be 1.
498
00:22:20,550 --> 00:22:22,800
When 'n' is 1, this
term is 5/12.
499
00:22:22,800 --> 00:22:24,570
When 'n' is 2, this is what?
500
00:22:24,570 --> 00:22:27,620
7 plus 17, 7/17.
501
00:22:27,620 --> 00:22:30,660
When 'n' is 3, this is 9/22.
502
00:22:30,660 --> 00:22:35,020
When 'n' is 4, this is 8
plus 3 is 11, over 27.
503
00:22:35,020 --> 00:22:37,070
In other words, what you're
saying is the infinite sum
504
00:22:37,070 --> 00:22:40,060
means to add up all
of these terms.
505
00:22:40,060 --> 00:22:44,080
The thing whose limit was 2/5
was the sequence of terms
506
00:22:44,080 --> 00:22:44,710
themselves.
507
00:22:44,710 --> 00:22:47,410
In other words, what we're
saying is that after a certain
508
00:22:47,410 --> 00:22:50,810
point, every one of these
terms behaves like 2/5.
509
00:22:50,810 --> 00:22:53,360
And what you're saying is
lookit, after a point, what
510
00:22:53,360 --> 00:22:56,610
you're really doing is
essentially adding on 2/5
511
00:22:56,610 --> 00:22:58,370
every time you add
on another term.
512
00:22:58,370 --> 00:23:02,230
And therefore, this sum can get
as large as you want, just
513
00:23:02,230 --> 00:23:03,840
by adding on enough terms.
514
00:23:03,840 --> 00:23:07,930
Again, observe the difference
between the partial sums and
515
00:23:07,930 --> 00:23:09,260
the terms themselves.
516
00:23:09,260 --> 00:23:11,230
The terms that you're
adding are
517
00:23:11,230 --> 00:23:13,430
approaching 2/5 as a limit.
518
00:23:13,430 --> 00:23:16,120
The thing that's becoming
infinite is the sequence of
519
00:23:16,120 --> 00:23:17,210
partial sums.
520
00:23:17,210 --> 00:23:20,300
Because what you're saying is to
get from one partial sum to
521
00:23:20,300 --> 00:23:22,150
the next, you're, roughly
speaking,
522
00:23:22,150 --> 00:23:24,940
adding on 2/5 each time.
523
00:23:24,940 --> 00:23:28,380
To generalize this, what we're
saying is if the sequence of
524
00:23:28,380 --> 00:23:32,220
partial sums converges, the
individual terms that you're
525
00:23:32,220 --> 00:23:35,210
adding must approach
0 in the limit.
526
00:23:35,210 --> 00:23:38,450
For if the limit of the 'a sub
n's as 'n' approaches infinity
527
00:23:38,450 --> 00:23:43,500
is 'L', where 'L' is not 0, then
beyond a certain term,
528
00:23:43,500 --> 00:23:45,820
the sum of the 'a sub
n's behaves like
529
00:23:45,820 --> 00:23:47,280
the sum of the 'L's.
530
00:23:47,280 --> 00:23:50,250
And what you're saying is if 'L'
is non zero, by adding on
531
00:23:50,250 --> 00:23:53,330
enough of these fixed 'L's, you
can make the sum as large
532
00:23:53,330 --> 00:23:55,380
as you wish.
533
00:23:55,380 --> 00:23:58,550
In other words, then, a sort
of negative test is that if
534
00:23:58,550 --> 00:24:01,520
you know that the series
converges, then the terms that
535
00:24:01,520 --> 00:24:04,520
you're adding on must approach
0 in the limit.
536
00:24:04,520 --> 00:24:07,650
Unfortunately, by the way,
the converse is not true.
537
00:24:07,650 --> 00:24:10,790
Namely, if you know that the
terms that you're adding on go
538
00:24:10,790 --> 00:24:15,680
to 0, you cannot conclude that
their sum is finite.
539
00:24:15,680 --> 00:24:19,230
Again, it's our old friend
of infinity times 0.
540
00:24:19,230 --> 00:24:25,140
You see, as these terms approach
0, when you start to
541
00:24:25,140 --> 00:24:27,840
add them up, it may be
that they're not
542
00:24:27,840 --> 00:24:29,030
going to 0 fast enough.
543
00:24:29,030 --> 00:24:31,160
In other words, notice that the
terms are getting small,
544
00:24:31,160 --> 00:24:33,500
but you're also adding more
and more of them.
545
00:24:33,500 --> 00:24:34,870
You see, what I wrote
here is what?
546
00:24:34,870 --> 00:24:37,630
On the other hand, the limit of
'a sub n' as 'n' approaches
547
00:24:37,630 --> 00:24:40,840
infinity equals 0 is not
enough to guarantee the
548
00:24:40,840 --> 00:24:43,460
convergence of this
particular sum.
549
00:24:43,460 --> 00:24:46,350
In fact, a trivial example to
show this is look at the
550
00:24:46,350 --> 00:24:48,440
following contrived example.
551
00:24:48,440 --> 00:24:50,970
Start out with the first
number being 1.
552
00:24:50,970 --> 00:24:57,720
Then take 1/2 twice, 1/3 three
times, 1/4 four times, 1/5
553
00:24:57,720 --> 00:25:00,520
five times, 1/6 six times.
554
00:25:00,520 --> 00:25:02,830
Form the n-th partial sum.
555
00:25:02,830 --> 00:25:06,100
Lookit, is it clear that the
terms that are going into your
556
00:25:06,100 --> 00:25:09,030
sum are approaching
0 in the limit?
557
00:25:09,030 --> 00:25:12,590
You see, you have a one, then
there are halves, then thirds,
558
00:25:12,590 --> 00:25:15,060
then fourths, then fifths,
then sixths,
559
00:25:15,060 --> 00:25:16,180
sevenths, et cetera.
560
00:25:16,180 --> 00:25:17,900
The terms themselves
are getting
561
00:25:17,900 --> 00:25:19,780
arbitrarily close to 0.
562
00:25:19,780 --> 00:25:22,060
On the other hand, what
is the sum becoming?
563
00:25:22,060 --> 00:25:23,820
Well, this adds up to 1.
564
00:25:23,820 --> 00:25:25,290
This adds up to 1.
565
00:25:25,290 --> 00:25:28,090
This adds up to 1, and
this that up to 1.
566
00:25:28,090 --> 00:25:31,700
And in other words, by taking
enough terms, I can tack on as
567
00:25:31,700 --> 00:25:34,540
many ones is I want, and
ultimately, even though the
568
00:25:34,540 --> 00:25:37,390
terms become small, the
sum becomes large.
569
00:25:37,390 --> 00:25:40,520
In fact, it's precisely because
of this unpleasantness
570
00:25:40,520 --> 00:25:44,790
that we have to go into a rather
difficult lecture next
571
00:25:44,790 --> 00:25:48,610
time, talking about OK, how
then can you tell when an
572
00:25:48,610 --> 00:25:51,840
infinite sum converges
to a finite limit and
573
00:25:51,840 --> 00:25:53,260
when doesn't it?
574
00:25:53,260 --> 00:25:54,430
At any rate, that's what
I said we're going to
575
00:25:54,430 --> 00:25:56,050
talk about next time.
576
00:25:56,050 --> 00:25:58,660
As far as today's lesson is
concerned, I hope that we've
577
00:25:58,660 --> 00:26:00,710
straightened out the difference
between a sequence
578
00:26:00,710 --> 00:26:04,100
and the series, partial sums
and the terms being added.
579
00:26:04,100 --> 00:26:06,590
And in the hopes that we've done
that, let me say, until
580
00:26:06,590 --> 00:26:07,840
next time, goodbye.
581
00:26:07,840 --> 00:26:09,970
582
00:26:09,970 --> 00:26:13,170
Funding for the publication of
this video was provided by the
583
00:26:13,170 --> 00:26:17,220
Gabriella and Paul Rosenbaum
Foundation.
584
00:26:17,220 --> 00:26:21,390
Help OCW continue to provide
free and open access to MIT
585
00:26:21,390 --> 00:26:25,590
courses by making a donation
at ocw.mit.edu/donate.
586
00:26:25,590 --> 00:26:30,335