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PROFESSOR: Hi.
00:00:33.050 --> 00:00:37.360
Last time, we had discussed
series and sequences.
00:00:37.360 --> 00:00:41.010
Today, we're going to turn our
attention to a rather special
00:00:41.010 --> 00:00:45.520
situation, a situation in which
every term in our series
00:00:45.520 --> 00:00:46.720
is positive.
00:00:46.720 --> 00:00:49.630
For this reason, I have entitled
today's lecture
00:00:49.630 --> 00:00:51.160
'Positive Series'.
00:00:51.160 --> 00:00:54.760
Before we can do full justice
to positive series, however,
00:00:54.760 --> 00:00:58.770
there are a few topics that we
must discuss as preliminaries.
00:00:58.770 --> 00:01:02.710
The first of these is simply
the process of ordering.
00:01:02.710 --> 00:01:05.290
Now this is a rather
strange situation.
00:01:05.290 --> 00:01:07.550
Because in the finite
case, it turns out
00:01:07.550 --> 00:01:08.670
to be rather trivial.
00:01:08.670 --> 00:01:12.550
By way of illustration, let's
suppose the set 'S' consists
00:01:12.550 --> 00:01:16.860
of the numbers 11,
8, 9, 7, and 10.
00:01:16.860 --> 00:01:18.510
Now there are many ways
in which we could
00:01:18.510 --> 00:01:19.610
order the set 'S'.
00:01:19.610 --> 00:01:22.530
We can arrange them so that any
one of these five elements
00:01:22.530 --> 00:01:24.340
comes first, any one
of the remaining
00:01:24.340 --> 00:01:25.790
four second, et cetera.
00:01:25.790 --> 00:01:28.270
But suppose that we want
to arrange these
00:01:28.270 --> 00:01:29.940
according to size.
00:01:29.940 --> 00:01:33.990
Observe that there is a rather
straightforward, shall we say,
00:01:33.990 --> 00:01:37.240
binary technique, whereby we
can order these elements.
00:01:37.240 --> 00:01:40.090
By binary, I mean, let's look
at these two at the time.
00:01:40.090 --> 00:01:43.020
We look at 11 and 8, and we
throw away the larger of the
00:01:43.020 --> 00:01:44.220
two, which is 11.
00:01:44.220 --> 00:01:47.020
Then we compare 8 with
9, throwaway 9
00:01:47.020 --> 00:01:48.180
because that's bigger.
00:01:48.180 --> 00:01:51.520
Compare 8 and 7, throw away 8.
00:01:51.520 --> 00:01:55.240
Compare 7 and 10,
throw away 10.
00:01:55.240 --> 00:01:59.685
The survivor, being 7, is
therefore the least member of
00:01:59.685 --> 00:02:00.790
our collection.
00:02:00.790 --> 00:02:04.810
In a similar way, we can delete
7 and start looking for
00:02:04.810 --> 00:02:06.770
the next number of our set.
00:02:06.770 --> 00:02:10.449
And in this way, eventually
order the elements of 'S'
00:02:10.449 --> 00:02:11.390
according to size--
00:02:11.390 --> 00:02:14.080
7, 8, 9, 10, 11.
00:02:14.080 --> 00:02:14.690
OK.
00:02:14.690 --> 00:02:20.690
Clearly, 7 is the smallest
element of 'S' and 11 is the
00:02:20.690 --> 00:02:22.860
largest element of S.
00:02:22.860 --> 00:02:27.210
And in terms of nomenclature, we
say that 7 is a lower bound
00:02:27.210 --> 00:02:30.840
for 'S', 11 is an upper
bound for 'S'.
00:02:30.840 --> 00:02:34.070
You see, in terms of a picture,
all we're saying is
00:02:34.070 --> 00:02:37.160
that when the elements of 'S'
are ordered according to size,
00:02:37.160 --> 00:02:41.220
7 is the furthest to
the left, 11 is the
00:02:41.220 --> 00:02:43.040
furthest to the right.
00:02:43.040 --> 00:02:46.490
We could even talk more about
the nomenclature by saying--
00:02:46.490 --> 00:02:48.210
and this sounds like
a tongue twister.
00:02:48.210 --> 00:02:51.290
This is why I had you read this
material first in the
00:02:51.290 --> 00:02:53.660
last assignment and the
supplementary notes, so that
00:02:53.660 --> 00:02:55.990
part of this will at least
seem like a review.
00:02:55.990 --> 00:02:59.740
Observe that 7 is called the
greatest lower bound for 'S',
00:02:59.740 --> 00:03:04.570
simply because any number larger
than 7 cannot be a
00:03:04.570 --> 00:03:08.350
lower bound for 'S', simply
because 7 would be smaller
00:03:08.350 --> 00:03:09.880
than that particular number.
00:03:09.880 --> 00:03:13.690
And in a similar way, 11 is
called the least upper bound
00:03:13.690 --> 00:03:17.650
for 'S', because any number
smaller than 11 would be
00:03:17.650 --> 00:03:20.520
exceeded by 11, and hence
could not be an
00:03:20.520 --> 00:03:22.260
upper bound for 'S'.
00:03:22.260 --> 00:03:24.110
Now the interesting
point is this.
00:03:24.110 --> 00:03:27.000
Hopefully, at this stage of the
game, you listened to what
00:03:27.000 --> 00:03:30.810
I've had to say, and you say,
my golly, this is trivial.
00:03:30.810 --> 00:03:32.720
And the answer is,
it is trivial.
00:03:32.720 --> 00:03:36.500
But remember what our main
concern was in our last
00:03:36.500 --> 00:03:40.460
lecture when we introduced the
concept of infinite series and
00:03:40.460 --> 00:03:41.490
sequences--
00:03:41.490 --> 00:03:45.740
that many things that were
trivial for the finite case
00:03:45.740 --> 00:03:49.800
became rather serious dilemmas
for the infinite case.
00:03:49.800 --> 00:03:52.960
In other words, my claim is
that these results are far
00:03:52.960 --> 00:03:55.260
more subtle for infinite sets.
00:03:55.260 --> 00:03:57.160
And I think the best
way to do this is
00:03:57.160 --> 00:03:59.000
by means of an example.
00:03:59.000 --> 00:04:03.560
See now, let 'S' be the set of
numbers where the n-th number
00:04:03.560 --> 00:04:05.860
is 'n' over 'n + 1'.
00:04:05.860 --> 00:04:08.480
In other words, the first member
of 'S' will be 1/2, the
00:04:08.480 --> 00:04:13.300
second member, 2/3, the third
member, 3/4, et cetera.
00:04:13.300 --> 00:04:16.050
Now, let's take look to see
what the least upper
00:04:16.050 --> 00:04:17.350
bound for 'S' is.
00:04:17.350 --> 00:04:20.500
And sparing you the details, I
think you can observe at this
00:04:20.500 --> 00:04:24.070
stage of the game, especially
based on the homework of the
00:04:24.070 --> 00:04:30.390
last unit, that the limit of
the sequence 'S' is 1.
00:04:30.390 --> 00:04:34.220
In fact, 1 is the smallest
number which exceeds every
00:04:34.220 --> 00:04:35.690
member in this collection.
00:04:35.690 --> 00:04:38.740
In other words, 1 is the least
upper bound for 'S'.
00:04:38.740 --> 00:04:41.945
But observe the interesting
case that here--
00:04:41.945 --> 00:04:43.820
and by the way, notice the
abbreviation that we use in
00:04:43.820 --> 00:04:46.700
our notes. 'LUB', least
upper bound.
00:04:46.700 --> 00:04:48.890
'GLB', greatest lower bound.
00:04:48.890 --> 00:04:51.730
But 1 is the least upper
bound for 'S'.
00:04:51.730 --> 00:04:55.740
Yet the fact remains that the
least upper bound is not a
00:04:55.740 --> 00:04:57.160
member of 'S' itself.
00:04:57.160 --> 00:05:00.960
In other words, there is no
number 'n' such that 'n' over
00:05:00.960 --> 00:05:03.470
'n + 1' is equal to 1.
00:05:03.470 --> 00:05:06.320
You see, notice that as these
numbers increase, they get
00:05:06.320 --> 00:05:09.620
bounded by 1, but 1 is
not a member of 'S'.
00:05:09.620 --> 00:05:12.380
In other words, here's an
example where the least upper
00:05:12.380 --> 00:05:16.110
bound of a set does not have
to be a member of the set.
00:05:16.110 --> 00:05:18.740
And a companion to this would
be an example where the
00:05:18.740 --> 00:05:21.950
greatest lower bound is not
a member of the set.
00:05:21.950 --> 00:05:26.030
And to this end, simply let
the n-th member of 'S'
00:05:26.030 --> 00:05:29.220
arranged by sequence be '1/n'.
00:05:29.220 --> 00:05:31.000
The n-th member is '1/n'.
00:05:31.000 --> 00:05:33.020
Therefore, 'S' is what?
00:05:33.020 --> 00:05:37.590
The set consisting of 1,
1/2, 1/3, et cetera.
00:05:37.590 --> 00:05:41.550
Observe that as the terms go
further and further out, they
00:05:41.550 --> 00:05:44.650
get arbitrarily close
to 0 in size--
00:05:44.650 --> 00:05:47.310
every one of these terms, '1/n',
no matter how big 'n'
00:05:47.310 --> 00:05:48.890
is is greater than 0.
00:05:48.890 --> 00:05:52.150
In other words, then, observe
that 0 will be the greatest
00:05:52.150 --> 00:05:56.350
lower bound for 'S', but 0
is not a member of 'S'.
00:05:56.350 --> 00:05:59.690
In other words, it seems that
things which are quite trivial
00:05:59.690 --> 00:06:02.550
for finite collections have
certain degrees of
00:06:02.550 --> 00:06:04.870
sophistication for infinite
collections.
00:06:04.870 --> 00:06:08.440
So what we're going to do now
is to establish a few basic
00:06:08.440 --> 00:06:09.780
definitions.
00:06:09.780 --> 00:06:11.380
And we'll do it in a
rather formal way.
00:06:11.380 --> 00:06:13.790
Our first definition
is the following.
00:06:13.790 --> 00:06:17.000
Given the set of numbers, 'S',
'M' is called an upper bound
00:06:17.000 --> 00:06:21.020
for 'S', If 'M' is greater than
or equal to 'x' for all
00:06:21.020 --> 00:06:22.410
'x' in 'S'.
00:06:22.410 --> 00:06:26.230
In other words, if 'M' exceeds
every member of 'S', 'M' is
00:06:26.230 --> 00:06:29.920
called an upper bound for 'S'.
00:06:29.920 --> 00:06:32.590
As I say, these these
definitions are very
00:06:32.590 --> 00:06:34.140
straightforward.
00:06:34.140 --> 00:06:35.430
The companion--
00:06:35.430 --> 00:06:37.380
well, let's go one
step further.
00:06:37.380 --> 00:06:39.900
By the way, observe that I
have in the interest of
00:06:39.900 --> 00:06:43.100
brevity left out the
corresponding definitions for
00:06:43.100 --> 00:06:43.740
lower bounds.
00:06:43.740 --> 00:06:45.680
But they are quite analogous.
00:06:45.680 --> 00:06:49.760
In other words, a lower bound
for 'S' would be a number that
00:06:49.760 --> 00:06:53.080
was less than or equal to
each member in 'S'.
00:06:53.080 --> 00:06:56.830
At any rate, then, 'M' is called
a least upper bound for
00:06:56.830 --> 00:07:00.580
'S' if first of all, 'M' is
an upper bound for 'S'.
00:07:00.580 --> 00:07:05.010
And secondly, if 'L' is less
than 'M', 'L' is not an upper
00:07:05.010 --> 00:07:06.090
bound for 'S'.
00:07:06.090 --> 00:07:08.260
In other words, least upper
bound means what?
00:07:08.260 --> 00:07:12.190
Anything smaller cannot
be an upper bound.
00:07:12.190 --> 00:07:15.540
Notice in terms of our previous
remark that the least
00:07:15.540 --> 00:07:19.630
upper bound need not
belong to 'x'.
00:07:19.630 --> 00:07:21.860
The companion to this
would be what?
00:07:21.860 --> 00:07:25.620
A greatest lower bound would be
a number which is a lower
00:07:25.620 --> 00:07:28.790
bound such that anything
greater could
00:07:28.790 --> 00:07:30.470
not be a lower bound.
00:07:30.470 --> 00:07:34.950
Again, all of this is easier to
see in terms of a picture.
00:07:34.950 --> 00:07:39.220
Visualize 'S' as being this
interval with little 'm' and
00:07:39.220 --> 00:07:41.910
capital 'M' being the endpoints
of the interval.
00:07:41.910 --> 00:07:45.280
Observe that capital 'M' is
the least upper bound.
00:07:45.280 --> 00:07:47.900
Little 'm' is the greatest
lower bound.
00:07:47.900 --> 00:07:52.470
Anything smaller than little
'm' will be lower bound.
00:07:52.470 --> 00:07:56.690
Anything greater than capital
'M' will be an upper bound.
00:07:56.690 --> 00:08:00.770
And notice that nothing in the
set 'S' itself can be either
00:08:00.770 --> 00:08:03.140
an upper bound or
a lower bound.
00:08:03.140 --> 00:08:08.210
Because anything inside 'S'
appears to the right of little
00:08:08.210 --> 00:08:11.230
'm' and to the left
of capital 'M'.
00:08:11.230 --> 00:08:14.220
And one final definition
as a preliminary.
00:08:14.220 --> 00:08:18.410
A set, 'S', is called bounded if
it has both an upper and a
00:08:18.410 --> 00:08:19.790
lower bound.
00:08:19.790 --> 00:08:23.080
And the key property that
we have to keep track of
00:08:23.080 --> 00:08:24.110
throughout this--
00:08:24.110 --> 00:08:25.210
and we won't prove this.
00:08:25.210 --> 00:08:27.760
In other words, in more
rigorous advanced math
00:08:27.760 --> 00:08:30.730
courses, this is proven
as a theorem.
00:08:30.730 --> 00:08:33.739
For our purposes, it seems
self-evident enough so that
00:08:33.739 --> 00:08:35.100
we're willing to accept it.
00:08:35.100 --> 00:08:37.830
And so rather than to belabor
the point, let us just accept
00:08:37.830 --> 00:08:41.870
as a key property that every
bounded set of numbers has a
00:08:41.870 --> 00:08:44.360
greatest lower bound and
at least upper bound.
00:08:44.360 --> 00:08:48.140
In other words, if a set is
bounded, we can find a
00:08:48.140 --> 00:08:52.680
smallest upper bound and
a largest lower bound.
00:08:52.680 --> 00:08:55.390
And this completes the
first portion of
00:08:55.390 --> 00:08:57.550
our preliminary material.
00:08:57.550 --> 00:09:00.050
The next portion of our
preliminary material before
00:09:00.050 --> 00:09:04.690
studying positive series
involves what we mean by a
00:09:04.690 --> 00:09:09.010
monotonic non-decreasing
sequence.
00:09:09.010 --> 00:09:12.580
A sequence is called monotonic
non-decreasing--
00:09:12.580 --> 00:09:16.160
and if you don't frighten at
these words, it's almost
00:09:16.160 --> 00:09:18.100
self-evident what this
thing means.
00:09:18.100 --> 00:09:21.120
It means that no term can
be smaller than the
00:09:21.120 --> 00:09:22.580
one that came before.
00:09:22.580 --> 00:09:26.370
In other words, the n-th term
is less than or equal to the
00:09:26.370 --> 00:09:29.070
'n plus first term'
for each 'n'.
00:09:29.070 --> 00:09:32.070
And wording that more
explicitly, it says what?
00:09:32.070 --> 00:09:35.010
'a sub 1' is less than or equal
to 'a sub 2' is less
00:09:35.010 --> 00:09:37.380
than or equal to 'a
sub 3', et cetera.
00:09:37.380 --> 00:09:40.030
And I hope that it's clear by
this time that it's not
00:09:40.030 --> 00:09:42.560
self-evident that a sequence
has to have this.
00:09:42.560 --> 00:09:45.510
Remember the subscripts simply
tell you the order in which
00:09:45.510 --> 00:09:46.610
the terms appear.
00:09:46.610 --> 00:09:49.850
It has no bearing on the
size of the term.
00:09:49.850 --> 00:09:52.860
For example, in an arbitrary
sequence, recall that the
00:09:52.860 --> 00:09:56.100
second term can be smaller in
magnitude than the first term.
00:09:56.100 --> 00:10:00.540
However, if the terms are
non-decreasing sequentially
00:10:00.540 --> 00:10:03.350
this way, the sequence
is called monotonic
00:10:03.350 --> 00:10:04.560
non-decreasing.
00:10:04.560 --> 00:10:07.260
And the problem that comes up
is, or the important question
00:10:07.260 --> 00:10:11.130
that comes up is, what's so
important about monotonic
00:10:11.130 --> 00:10:13.000
non-decreasing sequences?
00:10:13.000 --> 00:10:15.730
And the answer is that for
such sequences, two
00:10:15.730 --> 00:10:17.530
possibilities exist.
00:10:17.530 --> 00:10:21.600
In other words, either the terms
can keep getting larger
00:10:21.600 --> 00:10:23.695
and larger without bound--
00:10:23.695 --> 00:10:24.850
see?
00:10:24.850 --> 00:10:26.940
In other words, that the
sequences 'a sub n' has no
00:10:26.940 --> 00:10:28.160
upper bound.
00:10:28.160 --> 00:10:31.140
In which case, we say that the
limit of 'a sub n' as 'n'
00:10:31.140 --> 00:10:34.070
approaches infinity
is infinity.
00:10:34.070 --> 00:10:36.320
And for example, the
ordinary counting
00:10:36.320 --> 00:10:37.950
sequence has this property.
00:10:37.950 --> 00:10:42.400
See, 1, 2, 3, 4, 5, et cetera,
is a monotonic
00:10:42.400 --> 00:10:43.565
non-decreasing sequence.
00:10:43.565 --> 00:10:45.930
In fact, it's monotonic
increasing.
00:10:45.930 --> 00:10:49.000
Every member of the sequence is
greater than the one that
00:10:49.000 --> 00:10:49.980
came before.
00:10:49.980 --> 00:10:52.440
But as you go further and
further out, the terms
00:10:52.440 --> 00:10:54.750
increase without upper bound.
00:10:54.750 --> 00:10:56.000
OK?
00:10:56.000 --> 00:11:00.170
Now what is the other
possibility for a monotonic
00:11:00.170 --> 00:11:01.980
non-decreasing sequence?
00:11:01.980 --> 00:11:06.450
After all, the opposite, or the
only other possibility, is
00:11:06.450 --> 00:11:09.520
that if the 'a sub n'-- if it's
false, that the sequence
00:11:09.520 --> 00:11:12.460
doesn't have an upper bound,
then of course it must have an
00:11:12.460 --> 00:11:12.960
upper bound.
00:11:12.960 --> 00:11:14.350
That's the second case.
00:11:14.350 --> 00:11:16.950
In other words, suppose the
sequence has an upper bound.
00:11:16.950 --> 00:11:19.710
Then the interesting point
is that the limit of this
00:11:19.710 --> 00:11:21.790
sequence exists.
00:11:21.790 --> 00:11:25.980
And not only does it exist, but
the limit itself is the
00:11:25.980 --> 00:11:28.500
least upper bound
of the sequence.
00:11:28.500 --> 00:11:32.060
In other words, in the case
where the sequence is
00:11:32.060 --> 00:11:35.680
non-decreasing, if
it's bounded--
00:11:35.680 --> 00:11:36.900
if it's bounded--
00:11:36.900 --> 00:11:39.180
the least upper bound
will be the limit.
00:11:39.180 --> 00:11:41.150
Now you see, here's where we
use that key property.
00:11:41.150 --> 00:11:45.020
Namely, if a sequence is
bounded, it must have a least
00:11:45.020 --> 00:11:45.990
upper bound.
00:11:45.990 --> 00:11:48.650
Let's call that least
upper bound 'L'.
00:11:48.650 --> 00:11:50.780
And by the way, you'll notice
I wrote the word proof in
00:11:50.780 --> 00:11:51.910
quotation marks.
00:11:51.910 --> 00:11:54.920
It's simply to indicate that I
prefer to give you a geometric
00:11:54.920 --> 00:11:57.630
proof here rather than
an analytic one.
00:11:57.630 --> 00:12:00.060
But the analytic proof
follows word for
00:12:00.060 --> 00:12:02.780
word from this picture.
00:12:02.780 --> 00:12:05.530
In other words, it just
translates in the usual way.
00:12:05.530 --> 00:12:08.750
And we'll drill on this with
the exercises and the notes
00:12:08.750 --> 00:12:09.770
and the textbook.
00:12:09.770 --> 00:12:11.690
But at any rate, the
idea is this.
00:12:11.690 --> 00:12:14.960
To prove that 'L' is a limit,
what must I do?
00:12:14.960 --> 00:12:18.700
I must show that if I choose
any epsilon greater than 0,
00:12:18.700 --> 00:12:22.860
that all the terms beyond a
certain one lie between 'L
00:12:22.860 --> 00:12:25.920
minus epsilon' and
'L plus epsilon'.
00:12:25.920 --> 00:12:27.320
Now here's the way this works.
00:12:27.320 --> 00:12:29.360
Let's see if we can just read
the diagram and get this thing
00:12:29.360 --> 00:12:30.390
very quickly.
00:12:30.390 --> 00:12:35.060
First of all, at least one term
in my sequence has to be
00:12:35.060 --> 00:12:37.590
between 'L minus epsilon'
and 'L'.
00:12:37.590 --> 00:12:39.730
And the reason for that
is simply this.
00:12:39.730 --> 00:12:41.800
Remember 'L' is a least
upper bound.
00:12:41.800 --> 00:12:44.890
Because 'L' is a least upper
bound, 'L minus epsilon'
00:12:44.890 --> 00:12:46.580
cannot be an upper bound.
00:12:46.580 --> 00:12:50.080
Now if no term can get beyond
'L minus epsilon', then
00:12:50.080 --> 00:12:53.320
certainly 'L minus epsilon'
would be an upper bound.
00:12:53.320 --> 00:12:55.920
The fact that 'L minus epsilon'
isn't an upper bound,
00:12:55.920 --> 00:13:00.770
therefore, means that at least
one 'a sub n', say 'a sub
00:13:00.770 --> 00:13:04.220
capital N', is in this
interval here.
00:13:04.220 --> 00:13:09.340
Notice also that because 'L' is
an upper bound, no 'a sub
00:13:09.340 --> 00:13:11.320
n' can get beyond here.
00:13:11.320 --> 00:13:13.840
In other words, no 'a
sub n' lies between
00:13:13.840 --> 00:13:15.700
'L' and 'L plus epsilon'.
00:13:15.700 --> 00:13:18.560
Because in that particular case,
if that were to happen,
00:13:18.560 --> 00:13:20.960
'L' couldn't be an
upper bound.
00:13:20.960 --> 00:13:22.720
OK, so far, so good.
00:13:22.720 --> 00:13:24.980
That follows just from
the definition of
00:13:24.980 --> 00:13:26.370
the least upper bound.
00:13:26.370 --> 00:13:30.320
Now, we use the fact that the
sequence is non-decreasing.
00:13:30.320 --> 00:13:31.660
And that says what?
00:13:31.660 --> 00:13:36.165
That if little 'n' is greater
than capital 'N', 'a sub
00:13:36.165 --> 00:13:39.430
little n' is greater than or
equal to 'a sub capital N'.
00:13:39.430 --> 00:13:42.360
In other words, what this means
is if you list the 'a
00:13:42.360 --> 00:13:46.740
sub n's along this line, as 'n'
increases, the terms move
00:13:46.740 --> 00:13:49.500
progressively from
left to right.
00:13:49.500 --> 00:13:56.680
In other words, notice then that
if 'n' is greater than or
00:13:56.680 --> 00:14:04.190
equal to capital 'N', 'L minus
epsilon' is less than 'a sub
00:14:04.190 --> 00:14:07.220
capital N', which in turn is
less than or equal to 'a sub
00:14:07.220 --> 00:14:10.220
little n', which in turn is
less than or equal to 'L'.
00:14:10.220 --> 00:14:12.900
And that's just another
geometric way of saying that
00:14:12.900 --> 00:14:16.050
'a sub n' is within
epsilon of 'L'.
00:14:16.050 --> 00:14:19.630
And that's exactly the
definition that the 'a sub n's
00:14:19.630 --> 00:14:21.250
converge to the limit 'L'.
00:14:21.250 --> 00:14:24.520
Now I went over this rather
quickly simply because this is
00:14:24.520 --> 00:14:25.840
done in the textbook.
00:14:25.840 --> 00:14:28.500
And all I'm trying to do with
this lecture is to give you a
00:14:28.500 --> 00:14:31.040
quick overview of
what's going on.
00:14:31.040 --> 00:14:33.970
Well, you see, we're now roughly
15 minutes into our
00:14:33.970 --> 00:14:37.460
lecture and we haven't come
to the main topic yet.
00:14:37.460 --> 00:14:40.410
My claim is that the main
topic works very, very
00:14:40.410 --> 00:14:44.140
smoothly once we understand
these preliminaries.
00:14:44.140 --> 00:14:47.600
The main topic, you see, is
called positive series.
00:14:47.600 --> 00:14:49.850
And the definition of a positive
series is just as the
00:14:49.850 --> 00:14:51.230
name implies.
00:14:51.230 --> 00:14:55.890
If each term in the series is
at least as great as 0-- in
00:14:55.890 --> 00:14:58.470
other words, if each term
is non-negative--
00:14:58.470 --> 00:15:01.080
then the series is
called positive.
00:15:01.080 --> 00:15:04.010
Now why are positive
series important?
00:15:04.010 --> 00:15:07.320
And why do they tie in with
our previous discussion?
00:15:07.320 --> 00:15:10.290
Well, let's answer the
last question.
00:15:10.290 --> 00:15:13.140
The reason they tie in with
our previous discussion is
00:15:13.140 --> 00:15:17.990
that we have already seen that
by the sum of a series, we
00:15:17.990 --> 00:15:20.820
mean the limit of the sequence
of partial sums.
00:15:20.820 --> 00:15:24.150
To go from one partial sum to
the next, you add on the next
00:15:24.150 --> 00:15:25.290
term in the series.
00:15:25.290 --> 00:15:28.430
If each of the 'a sub n's
is positive, or at least
00:15:28.430 --> 00:15:32.590
non-negative, notice then that
the sequence of partial sums
00:15:32.590 --> 00:15:34.860
is monotonic non-decreasing.
00:15:34.860 --> 00:15:35.670
Why?
00:15:35.670 --> 00:15:38.790
Because to go from the n-th
partial sums to the 'n plus
00:15:38.790 --> 00:15:43.460
first' partial sum, you add
on 'a sub 'n plus 1''.
00:15:43.460 --> 00:15:47.560
And since 'a sub 'n plus 1''
is at least as big as 0, it
00:15:47.560 --> 00:15:51.040
means that the 'n plus first'
partial sum can be no smaller
00:15:51.040 --> 00:15:52.650
than the n-th partial sum.
00:15:52.650 --> 00:15:57.860
In other words, therefore, if
the series summation 'n' goes
00:15:57.860 --> 00:16:01.780
from 1 to infinity, 'a sub n' is
a positive series, it must
00:16:01.780 --> 00:16:05.900
either diverge to infinity, or
else it converges to the limit
00:16:05.900 --> 00:16:09.400
'L', where 'L' is the least
upper bound for the sequence
00:16:09.400 --> 00:16:11.140
of partial sums.
00:16:11.140 --> 00:16:11.660
OK?
00:16:11.660 --> 00:16:15.470
Now qualitatively, that's
the end of story.
00:16:15.470 --> 00:16:16.820
In other words, we
now know what?
00:16:16.820 --> 00:16:20.380
For a positive series, it either
diverges to infinity,
00:16:20.380 --> 00:16:24.570
or else it converges
to a sum, a limit.
00:16:24.570 --> 00:16:25.820
And that limit is what?
00:16:25.820 --> 00:16:29.460
The least upper bound for the
sequence of partial sums.
00:16:29.460 --> 00:16:32.610
The problem is that
quantitatively, we would like
00:16:32.610 --> 00:16:36.590
to have some criteria for
determining whether a positive
00:16:36.590 --> 00:16:38.610
series falls into the
first category
00:16:38.610 --> 00:16:40.190
or the second category.
00:16:40.190 --> 00:16:42.950
Notice, how can we tell whether
the series converges
00:16:42.950 --> 00:16:44.900
or whether it diverges?
00:16:44.900 --> 00:16:47.670
And you see, the reading
material in the text for this
00:16:47.670 --> 00:16:52.360
assignment focuses attention
on three major tests.
00:16:52.360 --> 00:16:55.060
And these are the ones I'd like
to go over with you once
00:16:55.060 --> 00:16:57.080
lightly, so to speak.
00:16:57.080 --> 00:17:01.300
The first test is called
the comparison test.
00:17:01.300 --> 00:17:03.140
And it almost sounds
self-evident.
00:17:03.140 --> 00:17:05.869
I just want to outline
how these proofs go.
00:17:05.869 --> 00:17:08.869
Because I think that once you
hear these things spoken, as
00:17:08.869 --> 00:17:12.780
you read the material, the
formal proofs will fit into a
00:17:12.780 --> 00:17:16.010
pattern much more nicely than if
you haven't heard the stuff
00:17:16.010 --> 00:17:18.060
said out loud, you see.
00:17:18.060 --> 00:17:19.079
The idea is this.
00:17:19.079 --> 00:17:22.270
Let's suppose that we know that
summation 'n' goes from 1
00:17:22.270 --> 00:17:27.150
to infinity, 'C sub n', is a
convergent positive series.
00:17:27.150 --> 00:17:29.070
In other words, all of
these are positive,
00:17:29.070 --> 00:17:30.920
and the series converges.
00:17:30.920 --> 00:17:33.700
Suppose I now have another
sequence of numbers, 'u sub
00:17:33.700 --> 00:17:37.210
n', where each 'u sub
n' is positive--
00:17:37.210 --> 00:17:39.800
see, it's at least as big as 0--
but can be no bigger than
00:17:39.800 --> 00:17:42.170
'C sub n' for each 'n'.
00:17:42.170 --> 00:17:45.660
Then the statement is that the
series formed by adding up the
00:17:45.660 --> 00:17:48.080
'u sub n's must also converge.
00:17:48.080 --> 00:17:50.810
Notice what you're saying is,
here's a bunch of positive
00:17:50.810 --> 00:17:53.920
terms that can't
get too large.
00:17:53.920 --> 00:17:56.360
And these terms in magnitude
are less
00:17:56.360 --> 00:17:57.560
than or equal to these.
00:17:57.560 --> 00:17:59.760
Therefore, what you're saying is
that this sum can't get too
00:17:59.760 --> 00:18:01.030
large either.
00:18:01.030 --> 00:18:03.770
And if you want to verbalize
that so that it becomes more
00:18:03.770 --> 00:18:06.080
formal, the idea is this.
00:18:06.080 --> 00:18:11.700
Let 'T sub n' denote the n-th
partial sums of the series
00:18:11.700 --> 00:18:12.980
involving the 'C sub n's.
00:18:12.980 --> 00:18:17.220
In other words, let 'T sub n'
be 'C sub 1' plus et cetera,
00:18:17.220 --> 00:18:19.020
up to 'C sub n'.
00:18:19.020 --> 00:18:22.780
And let 'S sub n' be the n-th
term in the sequence of
00:18:22.780 --> 00:18:25.470
partial sums of the series
involving the 'u's.
00:18:25.470 --> 00:18:29.010
In other words, let 'S sub
n' be 'u1' plus et cetera
00:18:29.010 --> 00:18:31.080
up to 'u sub n'.
00:18:31.080 --> 00:18:32.630
Now the idea is-- lookit.
00:18:32.630 --> 00:18:35.870
We know that each of the 'u's
in magnitude is smaller than
00:18:35.870 --> 00:18:37.120
each of the 'C's.
00:18:37.120 --> 00:18:40.350
Consequently, the sum of all
the 'u's must be no greater
00:18:40.350 --> 00:18:42.520
than the sum of all the 'C's.
00:18:42.520 --> 00:18:46.460
In other words, n-th partial
sum, 'S sub n', is less than
00:18:46.460 --> 00:18:50.800
or equal to the partial sum,
'T sub n' for each 'n'.
00:18:50.800 --> 00:18:56.230
Now, by our previous result,
knowing that these series
00:18:56.230 --> 00:18:59.990
summation 'C sub n' converges,
it converges to its least
00:18:59.990 --> 00:19:02.140
upper bound of partial sums.
00:19:02.140 --> 00:19:05.450
In other words, the limit of
'T sub n' as 'n' approaches
00:19:05.450 --> 00:19:08.970
infinity is some number 'T',
where 'T' is the least upper
00:19:08.970 --> 00:19:12.680
bound of the sequence of partial
sums of the series.
00:19:12.680 --> 00:19:16.520
Well in particular, then, since
each 'S sub n' is less
00:19:16.520 --> 00:19:20.570
than or equal to 'T sub n', 'S
sub n' itself certainly cannot
00:19:20.570 --> 00:19:23.030
exceed the least upper
bound, namely 'T'.
00:19:23.030 --> 00:19:25.130
In other words, 'S sub n'
can be no bigger than
00:19:25.130 --> 00:19:26.760
'T' for each 'n'.
00:19:26.760 --> 00:19:28.700
Well, what does this mean?
00:19:28.700 --> 00:19:31.350
It means, then, that
the sequence 'S sub
00:19:31.350 --> 00:19:32.900
n' itself is bounded.
00:19:32.900 --> 00:19:34.990
Well, it's bounded.
00:19:34.990 --> 00:19:38.130
It's a monotonic non-decreasing
sequence.
00:19:38.130 --> 00:19:40.820
Therefore, its limit exists.
00:19:40.820 --> 00:19:43.840
Not only does it exist, but it's
the least upper bound of
00:19:43.840 --> 00:19:45.430
the sequence of partial sums.
00:19:45.430 --> 00:19:47.660
In other words, this proof
is given in the text.
00:19:47.660 --> 00:19:51.700
All I want you to see is that
step by step, what this proof
00:19:51.700 --> 00:19:55.780
really does is it compares
magnitudes of terms.
00:19:55.780 --> 00:19:59.430
In other words, if one batch of
terms can't get lodged, if
00:19:59.430 --> 00:20:01.220
term by term, everything--
00:20:01.220 --> 00:20:05.310
another sequence is less than
these terms, then that second
00:20:05.310 --> 00:20:07.830
sum can't get too
large either.
00:20:07.830 --> 00:20:10.080
And this is just a formalization
of that.
00:20:10.080 --> 00:20:13.590
There are a few notes that we
should make first of all.
00:20:13.590 --> 00:20:18.930
Namely, the condition that 'u
sub n' be between 0 and 'C sub
00:20:18.930 --> 00:20:23.980
n' for all 'n' can be weakened
to cover the case where this
00:20:23.980 --> 00:20:27.290
is true only beyond
a certain point.
00:20:27.290 --> 00:20:29.380
Look, let me show you
what I mean by this.
00:20:29.380 --> 00:20:36.030
Let's suppose I look at the
sequence 1 plus 1/2 plus 1/3--
00:20:36.030 --> 00:20:37.640
let's do at a different one.
00:20:37.640 --> 00:20:43.410
1 plus 1/2 plus 1/4 plus 1/8
plus 1/16, et cetera.
00:20:43.410 --> 00:20:45.880
In other words, the geometric
series each of
00:20:45.880 --> 00:20:47.810
whose terms is 1/2.
00:20:47.810 --> 00:20:50.950
This series we know
converges OK.
00:20:50.950 --> 00:20:53.700
Now what the comparison test
says is, suppose you have
00:20:53.700 --> 00:20:54.950
something like this--
00:20:54.950 --> 00:21:05.400
1 plus 1/3 plus 1/5 plus 1/9
plus 1/17, et cetera.
00:21:05.400 --> 00:21:07.980
See, notice that each of these
terms is less than the
00:21:07.980 --> 00:21:10.050
corresponding term here.
00:21:10.050 --> 00:21:13.380
Consequently, since these
terms add up to a finite
00:21:13.380 --> 00:21:16.210
amount, these terms
here must also add
00:21:16.210 --> 00:21:17.600
up to a finite amount.
00:21:17.600 --> 00:21:20.950
But suppose for the sake of
argument I decide to replace
00:21:20.950 --> 00:21:25.120
the first term here by
10 to the sixth.
00:21:25.120 --> 00:21:26.530
I'll make this a million.
00:21:26.530 --> 00:21:29.720
Now notice that this sum is
going to become much larger.
00:21:29.720 --> 00:21:33.680
But the point is when I change
this from a 1 to 10 to the
00:21:33.680 --> 00:21:37.230
sixth, even though I made the
sum larger, I didn't change
00:21:37.230 --> 00:21:38.620
the finiteness of it.
00:21:38.620 --> 00:21:41.800
In other words, if I replace the
first four terms here by
00:21:41.800 --> 00:21:45.150
fantastically large numbers and
then keep the rest of the
00:21:45.150 --> 00:21:49.470
series intact, sure, I've made
to sum very, very large.
00:21:49.470 --> 00:21:52.600
But I've only change it by a
finite amount, which will in
00:21:52.600 --> 00:21:54.840
effect, not change
the convergence.
00:21:54.840 --> 00:21:58.140
That's all I was saying over
here, that the comparison test
00:21:58.140 --> 00:21:59.550
really hinges on what?
00:21:59.550 --> 00:22:02.360
Beyond a certain term, you
could stop making the
00:22:02.360 --> 00:22:03.800
comparison.
00:22:03.800 --> 00:22:06.640
And the second observation is
the converse to what we're
00:22:06.640 --> 00:22:07.730
talking about.
00:22:07.730 --> 00:22:12.590
Namely, if we know that 'u sub
n' is at least as big as 'd
00:22:12.590 --> 00:22:16.730
sub n' for each 'n', where the
series summation 'n' goes from
00:22:16.730 --> 00:22:21.990
1 to infinity, 'd sub n' is a
positive divergent series,
00:22:21.990 --> 00:22:27.360
then this series, summation 'u
sub n', must also diverge
00:22:27.360 --> 00:22:31.510
since its convergence would
imply the convergence of this.
00:22:31.510 --> 00:22:35.980
In other words, notice that
by the comparison test, if
00:22:35.980 --> 00:22:41.290
summation 'u sub n' converged,
the 'd n's, being less than
00:22:41.290 --> 00:22:44.230
the 'u n's would mean that
summation 'dn' end would have
00:22:44.230 --> 00:22:45.220
to converge also.
00:22:45.220 --> 00:22:47.990
At any rate, these are
the two portions of
00:22:47.990 --> 00:22:49.760
the comparison test.
00:22:49.760 --> 00:22:52.620
And this is what goes into
the comparison test.
00:22:52.620 --> 00:22:54.540
Now the interesting thing,
or the draw back to the
00:22:54.540 --> 00:22:56.620
comparison test is
simply this--
00:22:56.620 --> 00:23:00.760
that 99 times out of 100, if
you can find a series to
00:23:00.760 --> 00:23:04.500
compare a given series with, you
probably would have known
00:23:04.500 --> 00:23:06.730
whether the given series
converged or diverged in the
00:23:06.730 --> 00:23:07.870
first place.
00:23:07.870 --> 00:23:10.050
In other words, somehow or
other, to find the right
00:23:10.050 --> 00:23:13.570
series to compare something with
is a rather subtle thing
00:23:13.570 --> 00:23:16.440
if you didn't already know the
right answer to the problem.
00:23:16.440 --> 00:23:19.370
Well, at any rate, what I'm
trying to drive at is that the
00:23:19.370 --> 00:23:25.170
comparison test has as one of
its features a proof for a
00:23:25.170 --> 00:23:28.525
more interesting test-- a test
that's far less intuitive--
00:23:28.525 --> 00:23:30.980
called the 'ratio test'.
00:23:30.980 --> 00:23:32.970
The ratio test says
the following.
00:23:32.970 --> 00:23:35.600
Let's suppose again you're
given a positive series.
00:23:35.600 --> 00:23:39.610
What you do now is form a
sequence whereby each term in
00:23:39.610 --> 00:23:42.200
the sequence is the
ratio between two
00:23:42.200 --> 00:23:44.190
terms in the series.
00:23:44.190 --> 00:23:48.150
In other words, what I do is I
form the sequence by taking
00:23:48.150 --> 00:23:51.050
the second term divided by the
first term, the third term
00:23:51.050 --> 00:23:53.910
divided by the second term, the
fourth term divided by the
00:23:53.910 --> 00:23:54.590
third term.
00:23:54.590 --> 00:23:58.100
Now let's call that general
term 'u sub n'.
00:23:58.100 --> 00:24:00.360
This seems a little bit
abstract for you.
00:24:00.360 --> 00:24:03.230
Let's look at a more
tangible example.
00:24:03.230 --> 00:24:06.380
Suppose I take the series
summation 'n' goes from 1 to
00:24:06.380 --> 00:24:10.210
infinity, '10 to the n'
over 'n factorial'.
00:24:10.210 --> 00:24:14.620
Notice that in this case, the
n-th term is '10 to the n'
00:24:14.620 --> 00:24:16.340
over 'n factorial'.
00:24:16.340 --> 00:24:20.910
The 'n plus first' term is '10
to the 'n plus 1'' over ''n
00:24:20.910 --> 00:24:22.770
plus 1' factorial'.
00:24:22.770 --> 00:24:26.830
So 'u sub n' is the ratio
of the 'n plus first'
00:24:26.830 --> 00:24:28.850
term to the n-th term.
00:24:28.850 --> 00:24:32.120
In other words, '10 to the 'n
plus 1'' over ''n plus 1'
00:24:32.120 --> 00:24:36.700
factorial' divided by '10 to
the n' over 'n factorial'.
00:24:36.700 --> 00:24:40.100
'10 to the 'n plus 1'' divided
by '10 to the n' is simply 10.
00:24:40.100 --> 00:24:43.740
And ''n plus 1' factorial'
divided by 'n factorial' is
00:24:43.740 --> 00:24:45.560
simply 'n plus 1'.
00:24:45.560 --> 00:24:48.220
In other words, observe the
structure of the factorials
00:24:48.220 --> 00:24:51.900
that you get from the n-th to
the 'n plus first' simply by
00:24:51.900 --> 00:24:54.200
multiplying by 'n plus 1'.
00:24:54.200 --> 00:24:56.350
Again, the computational
details will be
00:24:56.350 --> 00:24:58.870
left for the exercises.
00:24:58.870 --> 00:25:01.570
At any rate, then, in this
particular case, 'u sub n'
00:25:01.570 --> 00:25:03.950
would be '10 over 'n plus 1''.
00:25:03.950 --> 00:25:06.220
Now here's what the
ratio test says.
00:25:06.220 --> 00:25:09.480
Assuming that the limit 'u sub
n' as 'n' approaches infinity
00:25:09.480 --> 00:25:12.220
exists, call it 'rho'.
00:25:12.220 --> 00:25:17.660
Then, the series converges if
rho less than 1 and diverges
00:25:17.660 --> 00:25:19.490
if rho is greater than 1.
00:25:19.490 --> 00:25:22.750
And the test fails
if rho equals 1.
00:25:22.750 --> 00:25:26.830
In other words, if the terms get
progressively smaller, so
00:25:26.830 --> 00:25:30.040
that the ratio and the limit
stays less than 1, then the
00:25:30.040 --> 00:25:31.450
series converges.
00:25:31.450 --> 00:25:34.590
If on the other hand, the ratio
in the limit exceeds 1,
00:25:34.590 --> 00:25:37.170
that means the terms are getting
big fast enough so
00:25:37.170 --> 00:25:39.020
that the series diverges.
00:25:39.020 --> 00:25:42.430
Let me point out an important
observation here.
00:25:42.430 --> 00:25:46.310
Notice the difference between
the limit equaling 1 and each
00:25:46.310 --> 00:25:48.490
term of the sequence
being less than 1.
00:25:48.490 --> 00:25:51.030
In other words, notice that even
if 'u sub n' is less than
00:25:51.030 --> 00:25:55.970
1 for every 'n', rho
may still equal 1.
00:25:55.970 --> 00:25:59.350
For example, look at the terms
'n' over 'n plus 1''.
00:25:59.350 --> 00:26:02.980
For each 'n', 'n' over 'n
plus 1' is less than 1.
00:26:02.980 --> 00:26:05.810
Yet the limit as 'n' approaches
infinity is exactly
00:26:05.810 --> 00:26:07.360
equal to 1.
00:26:07.360 --> 00:26:10.810
Now again, the formal proof of
this is given in the book.
00:26:10.810 --> 00:26:13.160
But I thought if I just take
a few minutes to show you
00:26:13.160 --> 00:26:16.150
geometrically what's happening
here, you'll get a better
00:26:16.150 --> 00:26:19.060
picture to understand what's
happening in the text.
00:26:19.060 --> 00:26:22.170
Let's prove this in the case
that rho is less than 1.
00:26:22.170 --> 00:26:25.310
Pictorially, if rho is less than
1, that means there's a
00:26:25.310 --> 00:26:27.310
space between rho and 1.
00:26:27.310 --> 00:26:30.600
That means I can choose an
epsilon such that rho plus
00:26:30.600 --> 00:26:34.660
epsilon, which I'll call it 'r',
is a positive number, but
00:26:34.660 --> 00:26:36.500
still less than 1.
00:26:36.500 --> 00:26:39.370
Now, by definition of rho
being the limit of the
00:26:39.370 --> 00:26:43.470
sequence 'a sub 'n plus 1'' over
'a sub n', that means I
00:26:43.470 --> 00:26:47.350
can find the capital 'N' for
this given epsilon, such that
00:26:47.350 --> 00:26:51.820
whenever 'n' is greater than
capital 'N', that 'a sub 'n
00:26:51.820 --> 00:26:56.040
plus 1' over 'a sub n' is less
than rho plus epsilon.
00:26:56.040 --> 00:26:59.180
In other words, is less than
'r', where 'r' is some fixed
00:26:59.180 --> 00:27:01.380
number now less than 1.
00:27:01.380 --> 00:27:05.450
Now let me apply this to
successive values of 'n'.
00:27:05.450 --> 00:27:09.150
In other words, taking
'n' to be capital--
00:27:09.150 --> 00:27:13.010
see, looking at this thing here,
taking 'n' to be capital
00:27:13.010 --> 00:27:18.030
'N', I have 'a' over capital 'N
plus 1', 'a sub capital 'N
00:27:18.030 --> 00:27:20.990
plus 1'' over 'a sub N'
is less than 'r'.
00:27:20.990 --> 00:27:24.960
In other words, 'a sub capital
'N plus 1'' is less than 'r'
00:27:24.960 --> 00:27:26.500
times 'a sub N'.
00:27:26.500 --> 00:27:31.720
Similarly, 'a sub capital 'N
plus 2'' over 'a sub capital
00:27:31.720 --> 00:27:34.240
'N plus 1'' is also
less than 'r'.
00:27:34.240 --> 00:27:38.190
In other words, 'a sub capital
'N plus 2'' is less than 'r'
00:27:38.190 --> 00:27:41.920
times 'a sub capital
'N plus 1''.
00:27:41.920 --> 00:27:45.890
But 'a sub 'N plus 1'' in
turn is less than 'r'
00:27:45.890 --> 00:27:47.300
times 'a sub N'.
00:27:47.300 --> 00:27:51.290
Putting this in here, 'a sub
'N plus 2' is less than 'r
00:27:51.290 --> 00:27:53.240
squared' times 'a sub N'..
00:27:53.240 --> 00:27:57.170
At any rate, if I now sum these
inequalities, you see
00:27:57.170 --> 00:27:58.450
what I get is what?
00:27:58.450 --> 00:28:01.430
The limit as we go from 'n plus
1' to infinity-- in other
00:28:01.430 --> 00:28:03.880
words, the tail end of
this particular sum
00:28:03.880 --> 00:28:05.340
is less than what?
00:28:05.340 --> 00:28:08.310
Well, I can factor out the
'a sub N' over here.
00:28:08.310 --> 00:28:10.680
And what's left inside
is what?
00:28:10.680 --> 00:28:14.780
'r' plus 'r squared' plus
'r cubed', et cetera.
00:28:14.780 --> 00:28:19.550
But this particular series is a
convergent geometric series.
00:28:19.550 --> 00:28:22.720
In other words, this must
converge because this
00:28:22.720 --> 00:28:25.870
converges, and this is less
than, term by term,
00:28:25.870 --> 00:28:27.280
the terms over here.
00:28:27.280 --> 00:28:30.910
In other words, notice that the
proof of the ratio test
00:28:30.910 --> 00:28:34.690
hinges on knowing two things,
the comparison test and the
00:28:34.690 --> 00:28:37.110
convergence of a geometric
series.
00:28:37.110 --> 00:28:40.490
Again, the reason I go through
this as rapidly as I do is
00:28:40.490 --> 00:28:44.360
that every detail is done
magnificently in the textbook.
00:28:44.360 --> 00:28:47.640
All I want you to see here is
the overview of how these
00:28:47.640 --> 00:28:49.730
tests come about.
00:28:49.730 --> 00:28:54.060
Finally, we have another
powerful test called the
00:28:54.060 --> 00:28:55.360
'integral test'.
00:28:55.360 --> 00:28:58.740
And the integral test
essentially equates positive
00:28:58.740 --> 00:29:01.480
series with improper
integrals.
00:29:01.480 --> 00:29:05.500
By the way, I have presented the
material, so to speak, in
00:29:05.500 --> 00:29:08.640
the order of appearance
in the textbook.
00:29:08.640 --> 00:29:12.940
You see, the comparison test,
ratio test, and integral test
00:29:12.940 --> 00:29:15.690
are given in that order
in the text.
00:29:15.690 --> 00:29:17.760
And so, I kept the
same order here.
00:29:17.760 --> 00:29:20.470
However, it's interesting to
point out that the integral
00:29:20.470 --> 00:29:24.450
test, in a way, is a companion
of the comparison test.
00:29:24.450 --> 00:29:26.600
And let me show you first
of all what the
00:29:26.600 --> 00:29:28.130
integral test says.
00:29:28.130 --> 00:29:31.570
It says, and I've written out
the formal statements here.
00:29:31.570 --> 00:29:34.250
I'll show you pictorially what
this means in a moment.
00:29:34.250 --> 00:29:37.040
But it says, suppose there is
a decreasing continuous
00:29:37.040 --> 00:29:39.410
function, 'f of x'.
00:29:39.410 --> 00:29:40.760
Notice the word continuous
in here.
00:29:40.760 --> 00:29:43.610
That guarantees that we can
integrate 'f of x'.
00:29:43.610 --> 00:29:48.360
Such that 'f' evaluated at each
integral value of 'x',
00:29:48.360 --> 00:29:52.240
say 'x' equals 'n', is 'u sub
n', where 'u sub n' is the
00:29:52.240 --> 00:29:54.860
n-th term of the positive
series 'u1'
00:29:54.860 --> 00:29:56.650
plus 'u2', et cetera.
00:29:56.650 --> 00:29:59.830
Then, what the integral test
says is that the series
00:29:59.830 --> 00:30:03.760
summation 'n' goes from 1 to
infinity 'u sub n', and the
00:30:03.760 --> 00:30:08.220
integral 1 to infinity, ''f
of x' dx', either converge
00:30:08.220 --> 00:30:11.140
together or diverge together.
00:30:11.140 --> 00:30:13.420
In other words, we can test
a particular series for
00:30:13.420 --> 00:30:16.110
convergence by knowing
whether a particular
00:30:16.110 --> 00:30:18.200
improper integral converges.
00:30:18.200 --> 00:30:20.390
And to show you what this
thing means pictorially,
00:30:20.390 --> 00:30:21.860
simply observe this.
00:30:21.860 --> 00:30:26.220
See, what we're saying is,
suppose that when you plot the
00:30:26.220 --> 00:30:27.800
terms of the series--
00:30:27.800 --> 00:30:31.300
so, 'u1', 'u2', 'u3', 'u4'--
00:30:31.300 --> 00:30:34.800
that these happen to be the
integral values of a
00:30:34.800 --> 00:30:37.770
continuous curve, 'y' equals
'f of x', which not only
00:30:37.770 --> 00:30:41.060
passes through these points, but
is a continuous decreasing
00:30:41.060 --> 00:30:43.100
function as this happens.
00:30:43.100 --> 00:30:47.300
Then what the statement is is
that the sum of these lengths
00:30:47.300 --> 00:30:50.200
converges if and only
if the area under
00:30:50.200 --> 00:30:52.610
the curve is finite.
00:30:52.610 --> 00:30:54.200
And the proof go something
like this.
00:30:54.200 --> 00:30:56.530
It's a rather ingenious
type of thing.
00:30:56.530 --> 00:31:00.470
You see, notice that if this
height is 'u sub 1', and the
00:31:00.470 --> 00:31:04.070
base of the rectangle is 1,
notice that numerically, the
00:31:04.070 --> 00:31:06.600
area of the rectangle--
it's quite in general.
00:31:06.600 --> 00:31:11.000
If the base of a rectangle is 1,
numerically the area of the
00:31:11.000 --> 00:31:14.080
rectangle equals the height,
because the area is the base
00:31:14.080 --> 00:31:14.930
times the height.
00:31:14.930 --> 00:31:17.620
If the base is 1, the area
equals the height.
00:31:17.620 --> 00:31:19.510
So the idea is simply this.
00:31:19.510 --> 00:31:20.370
Lookit.
00:31:20.370 --> 00:31:23.060
Suppose, for example,
that we look at this
00:31:23.060 --> 00:31:24.630
diagram over here.
00:31:24.630 --> 00:31:28.490
Notice that in this diagram,
if we stop at n, the area
00:31:28.490 --> 00:31:32.500
under this curve is the integral
from 1 to 'n', or 1
00:31:32.500 --> 00:31:35.350
to 'n plus 1', because of how
these lines are drawn.
00:31:35.350 --> 00:31:40.320
See, notice that the first
height stops at the number 2.
00:31:40.320 --> 00:31:43.850
The second base stops at
number 3, et cetera.
00:31:43.850 --> 00:31:46.950
The idea is this, though, that
the area under the curve in
00:31:46.950 --> 00:31:51.130
general, from 1 to 'n plus 1',
is integral from 1 to 'n plus
00:31:51.130 --> 00:31:53.060
1', ''f of x' dx'.
00:31:53.060 --> 00:31:58.320
On the other hand, the area of
the rectangles are what?
00:31:58.320 --> 00:32:02.970
'u1' plus 'u2' plus
'u3', up to 'u n'.
00:32:02.970 --> 00:32:07.190
In other words, for any given
'n', 'u1' up to u n' is
00:32:07.190 --> 00:32:10.110
greater than or equal to this
particular integral.
00:32:10.110 --> 00:32:14.290
Consequently, taking the
limit as 'n' goes to
00:32:14.290 --> 00:32:16.220
infinity, we get what?
00:32:16.220 --> 00:32:20.990
The summation 'u n' is greater
than or equal to integral from
00:32:20.990 --> 00:32:23.380
one to infinity ''f of x' dx'.
00:32:23.380 --> 00:32:26.530
Consequently, if this integral
diverges, meaning that this
00:32:26.530 --> 00:32:30.030
gets very large, the fact that
this can be no less than this
00:32:30.030 --> 00:32:33.150
means that this too
must also diverge.
00:32:33.150 --> 00:32:37.340
Correspondingly, if we now do
the same problem, but draw the
00:32:37.340 --> 00:32:41.980
thing slightly differently,
notice that now in this
00:32:41.980 --> 00:32:47.770
particular picture, the area
under the curve is integral
00:32:47.770 --> 00:32:50.780
from 1 to 'n', ''f of x' dx'.
00:32:50.780 --> 00:32:53.400
On the other hand, the area of
the rectangles are what?
00:32:53.400 --> 00:32:56.840
It's 'u2' plus 'u3'
up to 'u n'.
00:32:56.840 --> 00:33:01.220
But now you see the rectangles
are inscribed under the curve.
00:33:01.220 --> 00:33:05.220
Consequently, 'u2' plus et
cetera, up to 'u n', is less
00:33:05.220 --> 00:33:06.860
than this integral.
00:33:06.860 --> 00:33:10.767
Therefore, if I add 'u1' onto
both sides, the sum 'u1', et
00:33:10.767 --> 00:33:14.920
cetera, up to 'u n', is less
than 'u1' plus integral 1 to
00:33:14.920 --> 00:33:17.090
'n', ''f of x' dx'.
00:33:17.090 --> 00:33:20.680
If I now take the limit as n
goes to infinity, you see this
00:33:20.680 --> 00:33:23.920
becomes summation un.
00:33:23.920 --> 00:33:26.330
n goes from 1 to infinity.
00:33:26.330 --> 00:33:32.880
This becomes integral from 1
to infinity, ''f of x' dx'.
00:33:32.880 --> 00:33:37.120
And therefore, if this now
converges, the sum on the
00:33:37.120 --> 00:33:38.630
right is finite.
00:33:38.630 --> 00:33:41.780
Since the sum on the left cannot
exceed the sum on the
00:33:41.780 --> 00:33:45.240
right, the sum on the left
must also be finite.
00:33:45.240 --> 00:33:48.960
Consequently, the convergence
of the integral implies the
00:33:48.960 --> 00:33:50.930
convergence of the series.
00:33:50.930 --> 00:33:54.320
Again, I apologize for doing
this this rapidly.
00:33:54.320 --> 00:33:57.780
All I wanted you to do was
to get an idea of what's
00:33:57.780 --> 00:33:58.510
happening here.
00:33:58.510 --> 00:34:01.760
Because as I say, the book is
magnificent in this section.
00:34:01.760 --> 00:34:03.960
The proofs are very well
self-contained.
00:34:03.960 --> 00:34:07.790
At any rate, this gives us three
rather powerful tests,
00:34:07.790 --> 00:34:11.190
which I will drill you on in
the exercises for testing
00:34:11.190 --> 00:34:13.350
convergence of positive
series.
00:34:13.350 --> 00:34:16.590
What we're going to do next time
is to come to grips with
00:34:16.590 --> 00:34:17.989
a much more serious problem.
00:34:17.989 --> 00:34:22.969
And that is, what do you do if
the series isn't positive?
00:34:22.969 --> 00:34:24.380
But that's another story.
00:34:24.380 --> 00:34:26.360
And so until next
time, good bye.
00:34:29.170 --> 00:34:32.370
Funding for the publication of
this video was provided by the
00:34:32.370 --> 00:34:36.420
Gabriella and Paul Rosenbaum
Foundation.
00:34:36.420 --> 00:34:40.590
Help OCW continue to provide
free and open access to MIT
00:34:40.590 --> 00:34:44.800
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