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PROFESSOR: Hi.
00:00:27.510 --> 00:00:30.770
Today we begin our final block
of material in this particular
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course, and it's the segment
entitled Infinite Series.
00:00:34.650 --> 00:00:37.510
And perhaps the best way to
motivate this rather difficult
00:00:37.510 --> 00:00:40.260
block of material is in
terms of the concept
00:00:40.260 --> 00:00:42.080
of many versus infinite.
00:00:42.080 --> 00:00:45.190
In many respects, this
particular block could've been
00:00:45.190 --> 00:00:46.860
given much earlier
in the course.
00:00:46.860 --> 00:00:50.140
But somehow or other, until we
have some sort of a feeling as
00:00:50.140 --> 00:00:53.760
to what infinity really means,
we have a maturity problem in
00:00:53.760 --> 00:00:55.910
trying to really grasp
the significance of
00:00:55.910 --> 00:00:56.850
what's going on.
00:00:56.850 --> 00:00:59.120
In fact, in a manner of
speaking, with all of this
00:00:59.120 --> 00:01:02.130
experience, there may be a
maturity problem in trying to
00:01:02.130 --> 00:01:03.770
grasp the fundamental ideas.
00:01:03.770 --> 00:01:07.000
What I shall do throughout the
material on this block is to
00:01:07.000 --> 00:01:10.180
utilize the lectures again to
make sure that the concepts
00:01:10.180 --> 00:01:13.720
become crystallized and use the
learning exercises plus
00:01:13.720 --> 00:01:17.140
the text plus supplements notes
to make sure that the
00:01:17.140 --> 00:01:20.750
details are taken care of
in adequate fashion.
00:01:20.750 --> 00:01:23.040
At any rate, I've entitle
today's lecture
00:01:23.040 --> 00:01:24.450
'Many Versus Infinite'.
00:01:24.450 --> 00:01:27.290
And I thought the best way to
get started on this was to
00:01:27.290 --> 00:01:29.990
think of a number that's very
easy to write in terms of
00:01:29.990 --> 00:01:31.790
exponential notation.
00:01:31.790 --> 00:01:35.560
Let capital 'N' be 10 to the
10 to the 10th power.
00:01:35.560 --> 00:01:37.910
10 to the 10, by the way,
is 10 billion, a 1
00:01:37.910 --> 00:01:39.270
followed by 10 zeroes.
00:01:39.270 --> 00:01:42.030
That's 10 to the 10-billionth
power.
00:01:42.030 --> 00:01:44.530
That, of course, means, if
written in place value, that
00:01:44.530 --> 00:01:48.200
would be a 1 followed by
10 billion zeroes.
00:01:48.200 --> 00:01:50.190
And for those of you who would
like an exercise in
00:01:50.190 --> 00:01:52.810
multiplication and long division
and you want to
00:01:52.810 --> 00:01:55.790
compute the number of seconds
in a year and what have you,
00:01:55.790 --> 00:01:57.970
it turns out without
too much difficulty
00:01:57.970 --> 00:01:59.360
that it can be shown.
00:01:59.360 --> 00:02:02.990
That to write 1 billion zeroes
at the rate of one per second
00:02:02.990 --> 00:02:06.380
would take in the order of
magnitude of some 32 years.
00:02:06.380 --> 00:02:10.060
In other words, this number
capital 'N', roughly speaking,
00:02:10.060 --> 00:02:12.960
writing it in place value
notation at the rate of one
00:02:12.960 --> 00:02:16.640
digit per second would take
320 years to write.
00:02:16.640 --> 00:02:18.070
And you say so what?
00:02:18.070 --> 00:02:20.810
And the answer is, well, after
you've got out that far-- and
00:02:20.810 --> 00:02:22.560
by the way, this is crucial.
00:02:22.560 --> 00:02:25.380
320 years is a long time.
00:02:25.380 --> 00:02:26.230
I was going to say
it's a lifetime.
00:02:26.230 --> 00:02:27.610
It's more than the lifetime.
00:02:27.610 --> 00:02:29.860
It's a long time,
but it's finite.
00:02:29.860 --> 00:02:32.190
Eventually, the job could
be completed.
00:02:32.190 --> 00:02:34.710
But the interesting point is
that once it's completed, the
00:02:34.710 --> 00:02:38.400
next number in our system is
capital 'N plus 1', capital 'N
00:02:38.400 --> 00:02:42.920
plus 2', capital 'N plus 3',
where in a sense then, with
00:02:42.920 --> 00:02:46.140
'N' as a new reference point,
we're back to the beginning of
00:02:46.140 --> 00:02:47.030
our number system.
00:02:47.030 --> 00:02:50.740
In other words, granted that 'N'
is a fantastically large
00:02:50.740 --> 00:02:54.020
number, if you wanted to become
wealthy, to own 'N'
00:02:54.020 --> 00:02:56.810
dollars would more than
realize your dream.
00:02:56.810 --> 00:03:00.850
But if your aim was to own
infinitely much money, 'N'
00:03:00.850 --> 00:03:04.220
would be no closer than having
no money at all. 'N' is no
00:03:04.220 --> 00:03:06.520
nearer the end of a number
system than is
00:03:06.520 --> 00:03:07.990
the number 1 itself.
00:03:07.990 --> 00:03:11.020
There is the story that
signifies the difference
00:03:11.020 --> 00:03:12.670
between many and infinite.
00:03:12.670 --> 00:03:14.590
And to hammer this point
home, let me give
00:03:14.590 --> 00:03:16.600
you a few more examples.
00:03:16.600 --> 00:03:19.740
I cleverly call this additional
examples.
00:03:19.740 --> 00:03:22.720
We all know that there are just
as many odd numbers and
00:03:22.720 --> 00:03:23.640
even numbers, right?
00:03:23.640 --> 00:03:25.600
The odds and the
evens match up.
00:03:25.600 --> 00:03:27.630
Now, watch the following
little gimmick.
00:03:27.630 --> 00:03:30.450
Write the first two odd numbers,
then the first even
00:03:30.450 --> 00:03:33.490
number, the next two odd
numbers, then the next even
00:03:33.490 --> 00:03:35.570
number, the next two
odd numbers, than
00:03:35.570 --> 00:03:36.780
the next even number.
00:03:36.780 --> 00:03:39.080
And go on like this as
long as you want.
00:03:39.080 --> 00:03:43.510
And no matter where we stop,
even if we go to the 10 to the
00:03:43.510 --> 00:03:47.390
10 to the 10th term, no matter
what even number we stop at,
00:03:47.390 --> 00:03:50.470
there will always be twice as
many odd numbers written on
00:03:50.470 --> 00:03:53.450
the board as there would
be even numbers.
00:03:53.450 --> 00:03:57.090
In other words, even though in
the long run in terms of the
00:03:57.090 --> 00:04:00.070
infinity of each there are as
many odds and evens, if we
00:04:00.070 --> 00:04:03.650
stop this process at any finite
time no matter how far
00:04:03.650 --> 00:04:06.130
out, there will always
be twice as many odds
00:04:06.130 --> 00:04:07.130
as there are evens.
00:04:07.130 --> 00:04:10.290
In fact, if you want to compound
this little dilemma,
00:04:10.290 --> 00:04:12.680
write the first two evens, then
an odd, in other words,
00:04:12.680 --> 00:04:18.390
2, 4, 1, 6, 8, 3, 10, 12, 5, and
you can get twice as many
00:04:18.390 --> 00:04:20.490
evens as there are
odds, et cetera.
00:04:20.490 --> 00:04:22.890
And the whole argument
again hinges on what?
00:04:22.890 --> 00:04:26.560
Confusing the concept of
going out very far
00:04:26.560 --> 00:04:28.940
with going out endlessly.
00:04:28.940 --> 00:04:30.480
Oh, let me give you another
example or two.
00:04:30.480 --> 00:04:32.940
I just want to throw these
around so you at least get the
00:04:32.940 --> 00:04:36.850
mood created as to what we're
really dealing with right now.
00:04:36.850 --> 00:04:40.460
Let's take the endless sequence
of numbers, the sum,
00:04:40.460 --> 00:04:44.370
1 plus 'minus 1' plus 1
plus 'minus 1', and
00:04:44.370 --> 00:04:45.900
say let's go on forever.
00:04:45.900 --> 00:04:48.050
What will this sum be?
00:04:48.050 --> 00:04:49.760
Now, lookit, one
way of grouping
00:04:49.760 --> 00:04:51.750
these terms is in twos.
00:04:51.750 --> 00:04:54.550
In other words, we'll start with
the first two terms, the
00:04:54.550 --> 00:04:55.330
next two terms.
00:04:55.330 --> 00:04:59.460
In other words, we can write
this as 1 plus minus 1 plus 1
00:04:59.460 --> 00:05:00.700
plus minus 1.
00:05:00.700 --> 00:05:04.200
And writing it this way, we can
see that each term adds up
00:05:04.200 --> 00:05:07.210
to 0, and the infinite
sum would be 0.
00:05:07.210 --> 00:05:10.430
On the other hand, if we now
leave the first term alone and
00:05:10.430 --> 00:05:15.030
now start grouping the remaining
terms in twos, we
00:05:15.030 --> 00:05:17.440
find that the infinite
sum is 1.
00:05:17.440 --> 00:05:20.200
Now, we're not going to argue
that something is fishy here.
00:05:20.200 --> 00:05:21.340
We're not going to say
I wonder which
00:05:21.340 --> 00:05:22.170
is the right answer.
00:05:22.170 --> 00:05:26.790
What we have shown without fear
of contradiction is that
00:05:26.790 --> 00:05:29.600
the answer that you get when you
add infinitely many terms
00:05:29.600 --> 00:05:33.060
does depend on how you group
them, unlike the situation of
00:05:33.060 --> 00:05:35.560
what happens when you add
finitely many terms.
00:05:35.560 --> 00:05:38.730
In other words, notice the need
for order as well as the
00:05:38.730 --> 00:05:41.530
terms themselves when
you have a sum of
00:05:41.530 --> 00:05:42.960
infinitely many terms.
00:05:42.960 --> 00:05:47.000
And the key point is don't be
upset when you find out that
00:05:47.000 --> 00:05:48.480
your intuition is defied here.
00:05:48.480 --> 00:05:50.910
You say this doesn't
seem real to me.
00:05:50.910 --> 00:05:52.880
It seems intuitively false.
00:05:52.880 --> 00:05:56.110
The point is our intuition
is defied.
00:05:56.110 --> 00:05:56.850
Why?
00:05:56.850 --> 00:05:58.710
Because it doesn't apply.
00:05:58.710 --> 00:06:00.310
And why doesn't it apply?
00:06:00.310 --> 00:06:03.810
It doesn't apply because our
intuition is based on
00:06:03.810 --> 00:06:08.190
visualizing large but finite
amounts, not based on
00:06:08.190 --> 00:06:10.380
visualizing infinity.
00:06:10.380 --> 00:06:13.250
You see, all of these paradoxes
stem, because in our
00:06:13.250 --> 00:06:16.510
mind, we're trying to visualize
infinity as meaning
00:06:16.510 --> 00:06:18.010
the same as very large.
00:06:18.010 --> 00:06:20.470
Well, you know, now we come to
a very important crossroad.
00:06:20.470 --> 00:06:23.450
After all, if infinity is going
to be this difficult a
00:06:23.450 --> 00:06:26.770
concept to handle, let's get
rid of it the easy way.
00:06:26.770 --> 00:06:28.720
Let's refuse to study it.
00:06:28.720 --> 00:06:30.560
That's one way of solving
problems.
00:06:30.560 --> 00:06:33.140
It's what I call the right wing
conservative educational
00:06:33.140 --> 00:06:33.700
philosophy.
00:06:33.700 --> 00:06:36.060
If you don't like something,
throw it out.
00:06:36.060 --> 00:06:37.870
The only trouble
is we need it.
00:06:37.870 --> 00:06:40.160
For example, why
do we need it?
00:06:40.160 --> 00:06:41.380
See, why deal with
infinite sums?
00:06:41.380 --> 00:06:42.430
Well, because we need them.
00:06:42.430 --> 00:06:44.700
Among other places, we've
already used them.
00:06:44.700 --> 00:06:46.430
For example, in computing
areas.
00:06:46.430 --> 00:06:48.780
We've taken a limit as
'N' goes to infinity.
00:06:48.780 --> 00:06:53.240
Summation, 'k' goes from 1 to
'n', 'f of 'c sub k'' 'delta
00:06:53.240 --> 00:06:54.630
x', you see.
00:06:54.630 --> 00:06:55.840
And we need this limit.
00:06:55.840 --> 00:06:58.820
And so the question comes up,
how shall we add infinitely
00:06:58.820 --> 00:06:59.390
many terms?
00:06:59.390 --> 00:07:00.560
We have a choice now.
00:07:00.560 --> 00:07:02.480
We can throw the thing
out, but we don't
00:07:02.480 --> 00:07:03.180
want to throw it out.
00:07:03.180 --> 00:07:04.280
We need it.
00:07:04.280 --> 00:07:06.860
So the question is how shall we
add infinitely many terms?
00:07:06.860 --> 00:07:09.690
And even though we know that
our intuition can get us in
00:07:09.690 --> 00:07:13.260
trouble, we do have nothing
else to begin with.
00:07:13.260 --> 00:07:16.880
So we say OK, let's mimic what
happened in the finite case
00:07:16.880 --> 00:07:20.380
and see if we can't extend that
in a plausible way to
00:07:20.380 --> 00:07:21.720
cover the infinite case.
00:07:21.720 --> 00:07:24.900
Let me pick a particularly
straightforward example.
00:07:24.900 --> 00:07:27.840
Let's suppose I have the three
numbers which I'll call 'a sub
00:07:27.840 --> 00:07:31.730
1', 'a sub 2', and 'a sub 3',
where 'a sub 1' will be 1/2,
00:07:31.730 --> 00:07:35.310
'a sub 2' will be 1/4, and
'a sub 3' will be 1/8.
00:07:35.310 --> 00:07:37.640
In other words, just for reasons
of identification
00:07:37.640 --> 00:07:41.270
later on in what I'm going to be
doing, each term is half of
00:07:41.270 --> 00:07:43.190
the previous one.
00:07:43.190 --> 00:07:45.680
Now, I want to find the sum
of these three terms.
00:07:45.680 --> 00:07:49.320
I want to find 'a1' plus
'a2' plus 'a3'.
00:07:49.320 --> 00:07:51.750
Now, colloquially, we just say,
oh, that's 1/2 plus 1/4
00:07:51.750 --> 00:07:54.270
plus 1/8, and I'll
just add them up.
00:07:54.270 --> 00:07:55.650
But do you remember how
you learned to add?
00:07:55.650 --> 00:07:57.470
You may not have paid attention
to it, but you
00:07:57.470 --> 00:07:58.780
learned to add as a sequence.
00:07:58.780 --> 00:08:00.750
You said I'll add
the first one.
00:08:00.750 --> 00:08:04.030
Then the first plus the second
gives me a number.
00:08:04.030 --> 00:08:05.850
That's my second partial sum.
00:08:05.850 --> 00:08:07.500
Then I'll add on the
third number.
00:08:07.500 --> 00:08:09.830
That will give me my
third partial sum.
00:08:09.830 --> 00:08:11.940
Then I have no more
numbers to add.
00:08:11.940 --> 00:08:16.460
Consequently, my third partial
sum is by definition the sum
00:08:16.460 --> 00:08:17.840
of these three numbers.
00:08:17.840 --> 00:08:21.160
Writing it more symbolically,
we say lookit, the first
00:08:21.160 --> 00:08:23.720
partial sum, 's sub
1', is 1/2.
00:08:23.720 --> 00:08:26.430
The second partial sum
as 1/2 plus 1/4.
00:08:26.430 --> 00:08:27.990
Another way of saying
that is what?
00:08:27.990 --> 00:08:30.550
It's the first partial
sums plus the next
00:08:30.550 --> 00:08:31.860
term, which is 1/4.
00:08:31.860 --> 00:08:34.549
1/2 plus 1/4 is 3/4.
00:08:34.549 --> 00:08:38.049
Then we said OK, the third
partial sum is what we had
00:08:38.049 --> 00:08:42.820
before, namely, 3/4, plus the
next term, which was 1/8, and
00:08:42.820 --> 00:08:45.070
that gives rise to 7/8.
00:08:45.070 --> 00:08:49.320
In other words, we said let's
form 'a1', then 'a1 plus a2',
00:08:49.320 --> 00:08:51.640
'a1 plus a2 plus a3'.
00:08:51.640 --> 00:08:54.420
And when we finally finished
with our sequence of partial
00:08:54.420 --> 00:08:57.760
sums, the last partial
sum was the answer.
00:08:57.760 --> 00:09:01.330
And by the way, let me take time
out here to hit home at
00:09:01.330 --> 00:09:04.220
the most important point, the
point that I think is
00:09:04.220 --> 00:09:06.460
extremely crucial as a starting
point if we're going
00:09:06.460 --> 00:09:09.200
to understand what this whole
block is all about.
00:09:09.200 --> 00:09:12.135
It's to distinguish between
a series and a sequence.
00:09:12.135 --> 00:09:14.370
And I'll have much more to
say about this in the
00:09:14.370 --> 00:09:15.430
supplementary notes.
00:09:15.430 --> 00:09:17.500
But for now, think
of it this way.
00:09:17.500 --> 00:09:20.530
A series is a sum of terms.
00:09:20.530 --> 00:09:22.960
A sequence is just a
listing of terms.
00:09:22.960 --> 00:09:25.800
In other words, in this
particular problem, do not
00:09:25.800 --> 00:09:29.960
confuse the role of the 'a's
with the role of the 's's.
00:09:29.960 --> 00:09:33.240
Notice that the a's refer
to the sequence of
00:09:33.240 --> 00:09:34.850
numbers being added.
00:09:34.850 --> 00:09:36.600
In other words, the
'a's were what?
00:09:36.600 --> 00:09:39.170
They were 1/2, 1/4 and 1/8.
00:09:39.170 --> 00:09:41.440
These were the three numbers
being added.
00:09:41.440 --> 00:09:44.660
Notice that the 's's were
the partial sums.
00:09:44.660 --> 00:09:48.690
In other words, the partial sums
form the sequence 's1',
00:09:48.690 --> 00:09:50.430
's2', 's3'.
00:09:50.430 --> 00:09:52.780
And to refresh your memories
on this, that would be the
00:09:52.780 --> 00:09:53.770
sequence what?
00:09:53.770 --> 00:09:57.150
1/2, 3/4, 7/8.
00:09:57.150 --> 00:09:59.260
In other words, this was the sum
of the first number that
00:09:59.260 --> 00:10:00.060
you were adding.
00:10:00.060 --> 00:10:05.340
3/4 was the sum of first two,
and 7/8 was the sum of all
00:10:05.340 --> 00:10:06.240
three of them.
00:10:06.240 --> 00:10:10.330
And notice, by the way, the last
partial sum, 's sub 3',
00:10:10.330 --> 00:10:13.360
the sum was defined to
be the last partial
00:10:13.360 --> 00:10:14.910
sum, and that is what?
00:10:14.910 --> 00:10:15.790
The number--
00:10:15.790 --> 00:10:17.140
this is very, very crucial.
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
00:10:21.500 --> 00:10:24.480
one number, and that number
is called 7/8.
00:10:24.480 --> 00:10:26.280
OK, you see what we're
talking about now?
00:10:26.280 --> 00:10:27.960
We're looking at a
bunch of terms.
00:10:27.960 --> 00:10:30.390
We're adding them up, and
we see how the sum
00:10:30.390 --> 00:10:32.110
changes with each term.
00:10:32.110 --> 00:10:35.150
In fact, in terms of a very
trivial analogy, think of an
00:10:35.150 --> 00:10:37.450
adding machine.
00:10:37.450 --> 00:10:41.050
As you punch numbers in, the
's's are the sums that you see
00:10:41.050 --> 00:10:44.730
being read as your total sum,
whereas the a's are the
00:10:44.730 --> 00:10:48.450
individual numbers being punched
in to add up, OK?
00:10:48.450 --> 00:10:49.800
I hope that's a trivial
example.
00:10:49.800 --> 00:10:52.840
As I listen to myself saying it,
it sounds like I made it
00:10:52.840 --> 00:10:54.230
harder than it really is.
00:10:54.230 --> 00:10:57.230
At any rate, let's generalize
this particular problem.
00:10:57.230 --> 00:11:00.190
Let's suppose now instead of
wanting to add 1/2 plus 1/4
00:11:00.190 --> 00:11:04.320
plus 1/8, we want to add up
the first 'n' terms of the
00:11:04.320 --> 00:11:06.710
form 1/2 plus 1/4 plus 1/8.
00:11:06.710 --> 00:11:09.060
In other words, let the n-th
term that we're going to add,
00:11:09.060 --> 00:11:12.190
'a sub n', be '1 over
'2 to the n''.
00:11:12.190 --> 00:11:16.280
Then the sum, the n-th partial
sum here, the sum of these 'n'
00:11:16.280 --> 00:11:18.570
terms is, of course,
what? 'a1' plus, et
00:11:18.570 --> 00:11:19.880
cetera, 'a sub n'.
00:11:19.880 --> 00:11:23.950
That turns out to be 1/2 plus
1/4 plus, et cetera, '1 over
00:11:23.950 --> 00:11:25.680
'2 to the n''.
00:11:25.680 --> 00:11:29.670
And by the way, rather than
take time to develop this
00:11:29.670 --> 00:11:32.780
recipe over here, I thought you
might like to see another
00:11:32.780 --> 00:11:35.690
place that might be interesting
to review
00:11:35.690 --> 00:11:37.260
mathematical induction.
00:11:37.260 --> 00:11:40.280
If you'll bear with me and just
come back over here where
00:11:40.280 --> 00:11:44.660
we were computing these partial
sums, notice that in
00:11:44.660 --> 00:11:48.130
each of these partial sums,
notice that your denominator
00:11:48.130 --> 00:11:52.160
is always 2 raised to the same
power as this subscript.
00:11:52.160 --> 00:11:54.390
See, 2 the first power is 2.
00:11:54.390 --> 00:11:57.030
2 to the second power is 4.
00:11:57.030 --> 00:11:59.380
2 to the third power is 8.
00:11:59.380 --> 00:12:01.940
In other words, if your
subscript is n, your
00:12:01.940 --> 00:12:04.240
denominator is '2 to the n'.
00:12:04.240 --> 00:12:07.190
Notice that your numerator is
always one less than your
00:12:07.190 --> 00:12:08.080
denominator.
00:12:08.080 --> 00:12:11.180
In other words, if your
denominator is '2 to the n',
00:12:11.180 --> 00:12:14.060
the numerator is '2 to
the 'n minus 1''.
00:12:14.060 --> 00:12:17.570
And once we suspect this, this
particular result can be
00:12:17.570 --> 00:12:19.040
proven by induction.
00:12:19.040 --> 00:12:20.780
I won't take the time
to do this here.
00:12:20.780 --> 00:12:23.690
What I will take the time to
do is to observe that this
00:12:23.690 --> 00:12:27.250
particular sum can be written
more conveniently if we divide
00:12:27.250 --> 00:12:30.260
through by '2 to the n-th'
to get 1 minus
00:12:30.260 --> 00:12:31.990
'1 over '2 the n''.
00:12:31.990 --> 00:12:35.440
For example, suppose we wanted
to add up the 10 numbers.
00:12:35.440 --> 00:12:36.570
I say 10 numbers here.
00:12:36.570 --> 00:12:39.050
2 to the 10th is 1.024.
00:12:39.050 --> 00:12:43.190
But according to this recipe,
1/2 plus 1/4 plus 1/8 plus
00:12:43.190 --> 00:12:48.405
1/16 plus, et cetera, et cetera,
plus 1/1,024 would add
00:12:48.405 --> 00:12:49.390
up to be what?
00:12:49.390 --> 00:12:53.820
1 minus 1/1,024.
00:12:53.820 --> 00:12:59.300
In other words, this would be
1,023/1,024, which seems to be
00:12:59.300 --> 00:13:00.700
pretty close to 1.
00:13:00.700 --> 00:13:03.390
In fact, you can begin to
suspect that as 'n' gets
00:13:03.390 --> 00:13:06.660
arbitrarily large, 's sub
n' gets arbitrarily
00:13:06.660 --> 00:13:08.100
close to 1 in value.
00:13:08.100 --> 00:13:09.750
I'm just talking fairly
intuitively
00:13:09.750 --> 00:13:11.550
for the time being.
00:13:11.550 --> 00:13:14.460
But, you see, the major question
now is suppose you
00:13:14.460 --> 00:13:16.410
elect not to stop at that.
00:13:16.410 --> 00:13:17.860
And you see, this is
a very key point.
00:13:17.860 --> 00:13:21.810
We've already seen how the whole
world seems to change as
00:13:21.810 --> 00:13:25.720
soon as you say let's never
stop as opposed to saying
00:13:25.720 --> 00:13:27.650
let's go out as far
as you want.
00:13:27.650 --> 00:13:29.810
See, if we now say what
happens if you go on
00:13:29.810 --> 00:13:31.010
endlessly over here?
00:13:31.010 --> 00:13:34.190
Well, it becomes very natural
to say lookit, the n-th
00:13:34.190 --> 00:13:37.920
partial sum was 1 minus '1
over '2 to the n-th'.
00:13:37.920 --> 00:13:40.360
In the case where you were
adding up a finite number of
00:13:40.360 --> 00:13:44.040
terms, when you came to the last
partial sum, that was by
00:13:44.040 --> 00:13:45.550
definition your answer.
00:13:45.550 --> 00:13:47.970
Now, what we're saying is
lookit, because we have
00:13:47.970 --> 00:13:51.520
infinitely many terms to add,
there is no last partial sum.
00:13:51.520 --> 00:13:54.450
And so what we say is lookit,
instead of the last term,
00:13:54.450 --> 00:13:57.210
since there is no last term,
why don't we just take the
00:13:57.210 --> 00:14:01.160
limit of the n-th partial sum
as 'n' goes to infinity.
00:14:01.160 --> 00:14:03.605
In other words, in this
particular case, notice that
00:14:03.605 --> 00:14:07.220
as 'n' approaches infinity, 1
minus '1 over '2 to the n''
00:14:07.220 --> 00:14:10.910
approaches 1, and we then
define the infinite sum,
00:14:10.910 --> 00:14:12.010
meaning what?
00:14:12.010 --> 00:14:14.600
I write it this way: as sigma
'n' goes from 1 to infinity,
00:14:14.600 --> 00:14:16.050
'1 over '2 to the n''.
00:14:16.050 --> 00:14:17.150
It really means what?
00:14:17.150 --> 00:14:20.850
The limit as 'n' goes to
infinity: 1/2 plus 1/4 plus
00:14:20.850 --> 00:14:22.410
1/8 plus 1/16--
00:14:22.410 --> 00:14:24.190
endlessly--
00:14:24.190 --> 00:14:28.380
that that limit is 1, and we
define that to be the sum.
00:14:28.380 --> 00:14:31.730
And again, as I say, I'm going
to write that in greatly more
00:14:31.730 --> 00:14:34.160
detail in the notes, and
also we'll have many
00:14:34.160 --> 00:14:35.360
exercises on this.
00:14:35.360 --> 00:14:38.830
I just wanted you to see how we
get to infinite sums, which
00:14:38.830 --> 00:14:41.600
are called series by
generalizing what happens in
00:14:41.600 --> 00:14:42.930
the finite case.
00:14:42.930 --> 00:14:45.350
And because this may seem a
little vague to you, let me
00:14:45.350 --> 00:14:47.800
give you a pictorial
representation
00:14:47.800 --> 00:14:49.930
of this same thing.
00:14:49.930 --> 00:14:52.380
You see, what's happening
here is this.
00:14:52.380 --> 00:14:58.240
Draw a little circle around
1 of bandwidth epsilon.
00:14:58.240 --> 00:15:00.900
In other words, let's mark off
an interval epsilon on
00:15:00.900 --> 00:15:02.080
either side of 1.
00:15:02.080 --> 00:15:05.270
And let's call this point
here 1 minus epsilon.
00:15:05.270 --> 00:15:07.870
Let's call this point
here 1 plus epsilon.
00:15:07.870 --> 00:15:11.140
And what we're saying about
our partial sums is this.
00:15:11.140 --> 00:15:13.570
That when you start off and
you're adding up terms here,
00:15:13.570 --> 00:15:14.770
you have 1/2.
00:15:14.770 --> 00:15:18.720
1/2 plus 1/4 brings
you over to 3/4.
00:15:18.720 --> 00:15:21.420
The next possible sum
is 7/8, et cetera.
00:15:21.420 --> 00:15:24.870
And all we're saying is that
these terms get arbitrarily
00:15:24.870 --> 00:15:28.240
close to 1 in value, meaning
that after a while--
00:15:28.240 --> 00:15:30.450
and I'll define more rigorously
what after a while
00:15:30.450 --> 00:15:31.490
means in a moment--
00:15:31.490 --> 00:15:35.350
all of the 's sub n's are
within epsilon of 1.
00:15:35.350 --> 00:15:38.590
After a while, all of your
partial sums are in here.
00:15:38.590 --> 00:15:41.330
And what you mean by after a
while certainly depends on how
00:15:41.330 --> 00:15:42.260
big epsilon is.
00:15:42.260 --> 00:15:45.590
In other words, the smaller
the bandwidth you allow
00:15:45.590 --> 00:15:49.350
yourself, the more terms you
may have to take before you
00:15:49.350 --> 00:15:50.730
get within the tolerance limits
00:15:50.730 --> 00:15:52.270
that you allow yourself.
00:15:52.270 --> 00:15:55.500
In any event, going back to
something that we've been
00:15:55.500 --> 00:15:57.470
using for a long
time, our basic
00:15:57.470 --> 00:15:58.980
definition is the following.
00:15:58.980 --> 00:16:01.890
If you have an infinite
sequence, say, a collection of
00:16:01.890 --> 00:16:05.077
terms 'b sub n', in other words,
'b1', 'b2', 'b3', et
00:16:05.077 --> 00:16:09.900
cetera, without end, we say that
that sequence converges
00:16:09.900 --> 00:16:13.390
to the limit 'L' written the
limit of 'b sub n' as 'n'
00:16:13.390 --> 00:16:17.360
approaches infinity equals 'L',
if and only if for every
00:16:17.360 --> 00:16:21.510
epsilon greater than 0 we can
find a number 'N' which
00:16:21.510 --> 00:16:22.910
depends on epsilon--
00:16:22.910 --> 00:16:25.480
notice the notation here: 'N'
as a function of epsilon--
00:16:25.480 --> 00:16:29.280
such that whenever little 'n'
is greater than capital 'N',
00:16:29.280 --> 00:16:31.740
the absolute value of
'a sub n' minus 'L'
00:16:31.740 --> 00:16:33.050
is less than epsilon.
00:16:33.050 --> 00:16:35.755
And, you see, again, you may
wonder how in the world that
00:16:35.755 --> 00:16:36.920
you're going to remember this.
00:16:36.920 --> 00:16:39.690
If you memorize this, I
guarantee you, in two day's
00:16:39.690 --> 00:16:42.020
time, you'll have to
memorize it again.
00:16:42.020 --> 00:16:44.790
I also hope you have enough
faith in me to recognize I
00:16:44.790 --> 00:16:46.110
didn't memorize this.
00:16:46.110 --> 00:16:48.060
There is a feeling that
one gets for this.
00:16:48.060 --> 00:16:50.090
And let me give you what
that feeling is.
00:16:50.090 --> 00:16:53.140
Again, in terms of a picture,
what it means-- well, I'll
00:16:53.140 --> 00:16:55.720
change these to 'a's now because
that's the symbols
00:16:55.720 --> 00:16:58.170
that we've been using
before in terms of
00:16:58.170 --> 00:16:59.005
the sequence of terms.
00:16:59.005 --> 00:17:01.220
What we really mean-- and I
don't care what symbol you
00:17:01.220 --> 00:17:02.270
really use here--
00:17:02.270 --> 00:17:06.060
is if you want to talk about the
limit of 'a sub n' as 'n'
00:17:06.060 --> 00:17:08.930
approaches infinity, if that
limit equals 'L', what the
00:17:08.930 --> 00:17:10.940
rigorous definition
says is this.
00:17:10.940 --> 00:17:14.150
Draw yourself an interval
around 'L' of bandwidth
00:17:14.150 --> 00:17:15.740
epsilon, in other words,
from 'L minus
00:17:15.740 --> 00:17:18.089
epsilon' to 'L plus epsilon'.
00:17:18.089 --> 00:17:20.970
And what this thing says is that
beyond a certain term,
00:17:20.970 --> 00:17:24.250
say, the capital N-th term,
every term beyond this certain
00:17:24.250 --> 00:17:26.700
one is in here.
00:17:29.790 --> 00:17:35.240
Well, all 'a n's are in here if
'n' is sufficiently large.
00:17:35.240 --> 00:17:37.100
I don't know if you can read
that very well, but just
00:17:37.100 --> 00:17:38.020
listen to what I'm saying.
00:17:38.020 --> 00:17:42.060
All of the terms are in here if
'n' is sufficiently large.
00:17:42.060 --> 00:17:45.820
What this means again is that
to all intents and purposes,
00:17:45.820 --> 00:17:49.440
if you think of this bandwidth
as giving you a dot, see, a
00:17:49.440 --> 00:17:53.160
thick dot here where the
endpoints are 'L minus
00:17:53.160 --> 00:17:56.520
epsilon' and 'L plus epsilon',
what we're saying is lookit,
00:17:56.520 --> 00:17:59.600
after a certain term, the way
I've drawn here, after the
00:17:59.600 --> 00:18:02.290
fifth term, all the remaining
terms of my
00:18:02.290 --> 00:18:04.310
sequence are in here.
00:18:04.310 --> 00:18:07.740
By the way, notice the role
of the subscripts here.
00:18:07.740 --> 00:18:10.330
All the subscript tells
you is where the term
00:18:10.330 --> 00:18:12.550
appears in your sequence.
00:18:12.550 --> 00:18:15.300
For example, the third term in
your sequence could be a
00:18:15.300 --> 00:18:18.040
smaller number than the second
term of your sequence.
00:18:18.040 --> 00:18:21.210
Do not confuse the size of the
terms with the subscripts.
00:18:21.210 --> 00:18:24.170
The subscripts order the terms,
but the third term in
00:18:24.170 --> 00:18:27.790
the sequence can be less than
in size than the second term
00:18:27.790 --> 00:18:28.560
in the sequence.
00:18:28.560 --> 00:18:30.170
But again, I'll talk
about that in more
00:18:30.170 --> 00:18:31.280
detail in the notes.
00:18:31.280 --> 00:18:36.020
The point that I want you to
see is that in concept what
00:18:36.020 --> 00:18:37.840
limit does is the following.
00:18:37.840 --> 00:18:42.750
Limit is to an infinite sequence
as last term is to a
00:18:42.750 --> 00:18:44.200
finite sequence.
00:18:44.200 --> 00:18:48.320
In other words, a limit replaces
infinitely many
00:18:48.320 --> 00:18:52.720
points by a finite number
of points plus a dot.
00:18:52.720 --> 00:18:55.780
You see, going back to this
example here, how many 'a sub
00:18:55.780 --> 00:18:56.710
n's were there?
00:18:56.710 --> 00:18:58.760
Well, there were infinitely
many.
00:18:58.760 --> 00:19:01.530
Well, to keep track of these
infinitely many, what do I
00:19:01.530 --> 00:19:02.780
have to keep track of now?
00:19:02.780 --> 00:19:07.770
Well, in this diagram, the first
five 'a's plus this dot,
00:19:07.770 --> 00:19:12.290
because you see, every one of
my infinitely many 'a's past
00:19:12.290 --> 00:19:16.170
the fifth one is inside
this dot, you see?
00:19:16.170 --> 00:19:18.450
So in other words then,
what's happened?
00:19:18.450 --> 00:19:20.170
The thing that had to happen.
00:19:20.170 --> 00:19:22.670
We had to deal with infinite
sequences.
00:19:22.670 --> 00:19:26.200
We saw the big philosophic
difference between infinitely
00:19:26.200 --> 00:19:28.630
many and just large.
00:19:28.630 --> 00:19:33.550
And so our definition of limit
had to be such that we could
00:19:33.550 --> 00:19:37.930
reduce in a way that was
compatible with our intuition
00:19:37.930 --> 00:19:42.060
the concept of infinitely many
points to a finite number.
00:19:42.060 --> 00:19:45.430
Because, you see, as I'll show
you in the notes also, all of
00:19:45.430 --> 00:19:49.300
our arithmetic is geared for
just a finite number of
00:19:49.300 --> 00:19:50.610
operations.
00:19:50.610 --> 00:19:52.280
See, this is why
this definition
00:19:52.280 --> 00:19:53.920
of limit is so crucial.
00:19:53.920 --> 00:19:57.360
Again, you may notice, and I'll
remind you of this also
00:19:57.360 --> 00:20:00.960
in the exercises, that
structurally this definition
00:20:00.960 --> 00:20:04.040
of limit is the same as the
limit that we use when we
00:20:04.040 --> 00:20:07.000
talked about the limit of 'f of
x', as 'x' approaches 'a',
00:20:07.000 --> 00:20:07.880
equals 'L'.
00:20:07.880 --> 00:20:12.060
The absolute value signs have
the same properties as before.
00:20:12.060 --> 00:20:14.470
And by the way, before I go
on, let me just remind you
00:20:14.470 --> 00:20:17.810
again of one more thing while
I'm talking that way.
00:20:17.810 --> 00:20:20.950
Instead of memorizing this,
remember how you read this.
00:20:20.950 --> 00:20:24.180
The absolute value of 'a sub
n' minus 'L' is less than
00:20:24.180 --> 00:20:25.660
epsilon means what?
00:20:25.660 --> 00:20:28.820
That 'a sub n' is within
epsilon of 'L'.
00:20:28.820 --> 00:20:30.710
That's what we use
in our diagram.
00:20:30.710 --> 00:20:32.740
But it seems to me I forgot
to mention this.
00:20:32.740 --> 00:20:34.240
And I want you to
see that what?
00:20:34.240 --> 00:20:37.790
The key building block
analytically is the absolute
00:20:37.790 --> 00:20:40.690
value, and the meaning of
absolute value is the same
00:20:40.690 --> 00:20:45.210
here as it was in blocks one
and two of our course.
00:20:45.210 --> 00:20:49.010
So what I'm driving at is that
the same limit theorems that
00:20:49.010 --> 00:20:52.260
we've been able to use up
until now still apply.
00:20:52.260 --> 00:20:54.640
Oh, by means of an example.
00:20:54.640 --> 00:20:57.830
Suppose I have the limit as 'n'
approaches infinity, '2n
00:20:57.830 --> 00:21:01.110
plus 3' over '5n plus 7'.
00:21:01.110 --> 00:21:03.130
Notice that I can divide
numerator and denominator
00:21:03.130 --> 00:21:04.190
through by 'n'.
00:21:04.190 --> 00:21:07.100
If I do that, I have the limit
as 'n' approaches infinity.
00:21:07.100 --> 00:21:11.060
'2 plus '3/n'' over
'5 plus '7/n''.
00:21:11.060 --> 00:21:13.910
Now using the fact that the
limit of a sum is the sum of
00:21:13.910 --> 00:21:16.380
the limits, the limit of a
quotient is the quotient of
00:21:16.380 --> 00:21:21.320
the limits, the limit of '1/n'
as 'n' goes to infinity is 0.
00:21:21.320 --> 00:21:24.190
Notice that I can use the limit
theorems to conclude
00:21:24.190 --> 00:21:27.450
that the limit of this
particular sequence is 2/5.
00:21:27.450 --> 00:21:31.980
If I wanted to, the same ways
we did in block one, block
00:21:31.980 --> 00:21:35.710
two, where we're talking about
limits, given an epsilon, I
00:21:35.710 --> 00:21:39.360
can actually exhibit how far out
I have to go before each
00:21:39.360 --> 00:21:41.800
of the terms in this sequence
is within that
00:21:41.800 --> 00:21:44.180
given epsilon of 2/5.
00:21:44.180 --> 00:21:48.310
By the way, again to emphasize
once more, because this is so
00:21:48.310 --> 00:21:52.230
important, the difference
between an infinite sum and an
00:21:52.230 --> 00:21:55.560
infinite sequence, observe that
whereas the limit of the
00:21:55.560 --> 00:22:01.070
sequence of terms '2n plus 3'
over '5n plus 7' is 2/5, the
00:22:01.070 --> 00:22:07.360
infinite sum composed of the
terms of the form '2n plus 3'
00:22:07.360 --> 00:22:13.240
over '5n plus 7' is infinity
since after a while each term
00:22:13.240 --> 00:22:16.080
that you're adding here
behaves like 2/5.
00:22:16.080 --> 00:22:18.390
In other words, if you write
this thing out to see what
00:22:18.390 --> 00:22:20.550
this means, pick 'n' to be 1.
00:22:20.550 --> 00:22:22.800
When 'n' is 1, this
term is 5/12.
00:22:22.800 --> 00:22:24.570
When 'n' is 2, this is what?
00:22:24.570 --> 00:22:27.620
7 plus 17, 7/17.
00:22:27.620 --> 00:22:30.660
When 'n' is 3, this is 9/22.
00:22:30.660 --> 00:22:35.020
When 'n' is 4, this is 8
plus 3 is 11, over 27.
00:22:35.020 --> 00:22:37.070
In other words, what you're
saying is the infinite sum
00:22:37.070 --> 00:22:40.060
means to add up all
of these terms.
00:22:40.060 --> 00:22:44.080
The thing whose limit was 2/5
was the sequence of terms
00:22:44.080 --> 00:22:44.710
themselves.
00:22:44.710 --> 00:22:47.410
In other words, what we're
saying is that after a certain
00:22:47.410 --> 00:22:50.810
point, every one of these
terms behaves like 2/5.
00:22:50.810 --> 00:22:53.360
And what you're saying is
lookit, after a point, what
00:22:53.360 --> 00:22:56.610
you're really doing is
essentially adding on 2/5
00:22:56.610 --> 00:22:58.370
every time you add
on another term.
00:22:58.370 --> 00:23:02.230
And therefore, this sum can get
as large as you want, just
00:23:02.230 --> 00:23:03.840
by adding on enough terms.
00:23:03.840 --> 00:23:07.930
Again, observe the difference
between the partial sums and
00:23:07.930 --> 00:23:09.260
the terms themselves.
00:23:09.260 --> 00:23:11.230
The terms that you're
adding are
00:23:11.230 --> 00:23:13.430
approaching 2/5 as a limit.
00:23:13.430 --> 00:23:16.120
The thing that's becoming
infinite is the sequence of
00:23:16.120 --> 00:23:17.210
partial sums.
00:23:17.210 --> 00:23:20.300
Because what you're saying is to
get from one partial sum to
00:23:20.300 --> 00:23:22.150
the next, you're, roughly
speaking,
00:23:22.150 --> 00:23:24.940
adding on 2/5 each time.
00:23:24.940 --> 00:23:28.380
To generalize this, what we're
saying is if the sequence of
00:23:28.380 --> 00:23:32.220
partial sums converges, the
individual terms that you're
00:23:32.220 --> 00:23:35.210
adding must approach
0 in the limit.
00:23:35.210 --> 00:23:38.450
For if the limit of the 'a sub
n's as 'n' approaches infinity
00:23:38.450 --> 00:23:43.500
is 'L', where 'L' is not 0, then
beyond a certain term,
00:23:43.500 --> 00:23:45.820
the sum of the 'a sub
n's behaves like
00:23:45.820 --> 00:23:47.280
the sum of the 'L's.
00:23:47.280 --> 00:23:50.250
And what you're saying is if 'L'
is non zero, by adding on
00:23:50.250 --> 00:23:53.330
enough of these fixed 'L's, you
can make the sum as large
00:23:53.330 --> 00:23:55.380
as you wish.
00:23:55.380 --> 00:23:58.550
In other words, then, a sort
of negative test is that if
00:23:58.550 --> 00:24:01.520
you know that the series
converges, then the terms that
00:24:01.520 --> 00:24:04.520
you're adding on must approach
0 in the limit.
00:24:04.520 --> 00:24:07.650
Unfortunately, by the way,
the converse is not true.
00:24:07.650 --> 00:24:10.790
Namely, if you know that the
terms that you're adding on go
00:24:10.790 --> 00:24:15.680
to 0, you cannot conclude that
their sum is finite.
00:24:15.680 --> 00:24:19.230
Again, it's our old friend
of infinity times 0.
00:24:19.230 --> 00:24:25.140
You see, as these terms approach
0, when you start to
00:24:25.140 --> 00:24:27.840
add them up, it may be
that they're not
00:24:27.840 --> 00:24:29.030
going to 0 fast enough.
00:24:29.030 --> 00:24:31.160
In other words, notice that the
terms are getting small,
00:24:31.160 --> 00:24:33.500
but you're also adding more
and more of them.
00:24:33.500 --> 00:24:34.870
You see, what I wrote
here is what?
00:24:34.870 --> 00:24:37.630
On the other hand, the limit of
'a sub n' as 'n' approaches
00:24:37.630 --> 00:24:40.840
infinity equals 0 is not
enough to guarantee the
00:24:40.840 --> 00:24:43.460
convergence of this
particular sum.
00:24:43.460 --> 00:24:46.350
In fact, a trivial example to
show this is look at the
00:24:46.350 --> 00:24:48.440
following contrived example.
00:24:48.440 --> 00:24:50.970
Start out with the first
number being 1.
00:24:50.970 --> 00:24:57.720
Then take 1/2 twice, 1/3 three
times, 1/4 four times, 1/5
00:24:57.720 --> 00:25:00.520
five times, 1/6 six times.
00:25:00.520 --> 00:25:02.830
Form the n-th partial sum.
00:25:02.830 --> 00:25:06.100
Lookit, is it clear that the
terms that are going into your
00:25:06.100 --> 00:25:09.030
sum are approaching
0 in the limit?
00:25:09.030 --> 00:25:12.590
You see, you have a one, then
there are halves, then thirds,
00:25:12.590 --> 00:25:15.060
then fourths, then fifths,
then sixths,
00:25:15.060 --> 00:25:16.180
sevenths, et cetera.
00:25:16.180 --> 00:25:17.900
The terms themselves
are getting
00:25:17.900 --> 00:25:19.780
arbitrarily close to 0.
00:25:19.780 --> 00:25:22.060
On the other hand, what
is the sum becoming?
00:25:22.060 --> 00:25:23.820
Well, this adds up to 1.
00:25:23.820 --> 00:25:25.290
This adds up to 1.
00:25:25.290 --> 00:25:28.090
This adds up to 1, and
this that up to 1.
00:25:28.090 --> 00:25:31.700
And in other words, by taking
enough terms, I can tack on as
00:25:31.700 --> 00:25:34.540
many ones is I want, and
ultimately, even though the
00:25:34.540 --> 00:25:37.390
terms become small, the
sum becomes large.
00:25:37.390 --> 00:25:40.520
In fact, it's precisely because
of this unpleasantness
00:25:40.520 --> 00:25:44.790
that we have to go into a rather
difficult lecture next
00:25:44.790 --> 00:25:48.610
time, talking about OK, how
then can you tell when an
00:25:48.610 --> 00:25:51.840
infinite sum converges
to a finite limit and
00:25:51.840 --> 00:25:53.260
when doesn't it?
00:25:53.260 --> 00:25:54.430
At any rate, that's what
I said we're going to
00:25:54.430 --> 00:25:56.050
talk about next time.
00:25:56.050 --> 00:25:58.660
As far as today's lesson is
concerned, I hope that we've
00:25:58.660 --> 00:26:00.710
straightened out the difference
between a sequence
00:26:00.710 --> 00:26:04.100
and the series, partial sums
and the terms being added.
00:26:04.100 --> 00:26:06.590
And in the hopes that we've done
that, let me say, until
00:26:06.590 --> 00:26:07.840
next time, goodbye.
00:26:09.970 --> 00:26:13.170
Funding for the publication of
this video was provided by the
00:26:13.170 --> 00:26:17.220
Gabriella and Paul Rosenbaum
Foundation.
00:26:17.220 --> 00:26:21.390
Help OCW continue to provide
free and open access to MIT
00:26:21.390 --> 00:26:25.590
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at ocw.mit.edu/donate.