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HERBERT GROSS: Hi, last time I
left you dangling in suspense
00:00:35.290 --> 00:00:38.830
when I said what happens if the
terms in a series are not
00:00:38.830 --> 00:00:39.580
all positive?
00:00:39.580 --> 00:00:42.550
In other words, the trouble with
the last assignment was
00:00:42.550 --> 00:00:45.100
that we did quite a bit of work
and yet there was a very
00:00:45.100 --> 00:00:46.080
stringent condition.
00:00:46.080 --> 00:00:49.770
Namely, that every term that you
were adding happened to be
00:00:49.770 --> 00:00:50.790
a positive number.
00:00:50.790 --> 00:00:53.320
Now obviously, this need
not be the case.
00:00:53.320 --> 00:00:57.750
And the question is, can you
have convergence in a series
00:00:57.750 --> 00:01:00.300
in which the terms are
not all positive?
00:01:00.300 --> 00:01:03.100
And what does it mean, by the
way, if this is the case why
00:01:03.100 --> 00:01:06.980
we stressed the situation
of positive series?
00:01:06.980 --> 00:01:09.430
And this will be the aim of
today's lecture to straighten
00:01:09.430 --> 00:01:10.705
out both of these points.
00:01:10.705 --> 00:01:14.430
At any rate, I call today's
lesson 'Absolute Convergence',
00:01:14.430 --> 00:01:17.450
and I hope that the meaning of
this will become clear very
00:01:17.450 --> 00:01:18.750
soon as we go along.
00:01:18.750 --> 00:01:21.810
But to answer the first question
that we brought up,
00:01:21.810 --> 00:01:24.110
let's take a series
in which the
00:01:24.110 --> 00:01:25.950
terms are not all positive.
00:01:25.950 --> 00:01:29.450
Now, by the way, I will do more
general things in our
00:01:29.450 --> 00:01:31.470
supplementary notes
on this material.
00:01:31.470 --> 00:01:34.170
I felt though that for a
blackboard illustration, I
00:01:34.170 --> 00:01:37.110
should pick a relatively
straightforward example and
00:01:37.110 --> 00:01:38.720
not try to generalize it.
00:01:38.720 --> 00:01:46.140
Let's take the specific series 1
minus 1/2 plus 1/3 minus 1/4
00:01:46.140 --> 00:01:47.750
plus 1/5, et cetera.
00:01:47.750 --> 00:01:50.470
And how do we indicate the
n-th term in this case?
00:01:50.470 --> 00:01:53.400
Notice that the denominator
is 'n'.
00:01:53.400 --> 00:01:57.840
The numerator oscillates
between minus 1 and 1.
00:01:57.840 --> 00:02:01.720
And you see the mathematical
trick to alternate signs is to
00:02:01.720 --> 00:02:03.690
raise minus 1 to a power.
00:02:03.690 --> 00:02:07.290
You see minus 1 to an even
power will be positive 1.
00:02:07.290 --> 00:02:09.210
And to an odd power,
negative 1.
00:02:09.210 --> 00:02:12.860
The idea here is since for 'n'
equals 1 we want this to be
00:02:12.860 --> 00:02:15.370
positive, we tacked on
the plus 1 here.
00:02:15.370 --> 00:02:19.780
You see in that case, 'n' being
1, 'n plus 1' is 2.
00:02:19.780 --> 00:02:21.840
Minus 1 squared is 1.
00:02:21.840 --> 00:02:25.320
And at any rate, don't be
confused by this notation,
00:02:25.320 --> 00:02:28.700
it's simply a cute way all
of alternating signs.
00:02:28.700 --> 00:02:31.560
At any rate, what do we have
in this particular series?
00:02:31.560 --> 00:02:35.530
We have that the terms
alternate in sign.
00:02:35.530 --> 00:02:39.050
We also have that the n-th
term approaches 0.
00:02:39.050 --> 00:02:41.750
Namely, the numerator
alternates between
00:02:41.750 --> 00:02:43.220
plus and minus 1.
00:02:43.220 --> 00:02:44.990
The denominator is 'n'.
00:02:44.990 --> 00:02:50.710
So as 'n' increases, the terms
converge on 0 as a limit.
00:02:50.710 --> 00:02:54.470
And finally, the terms
keep decreasing
00:02:54.470 --> 00:02:56.650
monotonically in magnitude.
00:02:56.650 --> 00:02:59.620
In other words, forgetting about
the fact that a plus
00:02:59.620 --> 00:03:03.310
outranks a minus in terms of a
number line, notice that the
00:03:03.310 --> 00:03:06.930
size of 1/3 is less than
the size of 1/2.
00:03:06.930 --> 00:03:08.990
In other words, this particular
series, which is
00:03:08.990 --> 00:03:12.600
called an 'alternating series',
has three properties.
00:03:12.600 --> 00:03:14.680
It's called alternating
because the terms
00:03:14.680 --> 00:03:16.450
alternate in sign.
00:03:16.450 --> 00:03:18.080
The n-th term approaches 0.
00:03:18.080 --> 00:03:21.600
By the way, we saw in our first
lecture on series that
00:03:21.600 --> 00:03:24.980
the n-th term approaching
0 was necessary, but not
00:03:24.980 --> 00:03:27.850
sufficient for making
a series converge.
00:03:27.850 --> 00:03:32.120
However, what our claim is that
if in addition to this we
00:03:32.120 --> 00:03:35.140
know that the terms decrease
in magnitude, our claim is
00:03:35.140 --> 00:03:37.460
that the given series
will converge.
00:03:37.460 --> 00:03:40.770
In other words, what I intend
to show is that 1 minus 1/2
00:03:40.770 --> 00:03:43.670
plus 1/3, et cetera,
does converge.
00:03:43.670 --> 00:03:46.580
And the way I'm going to do
that is geometrically.
00:03:46.580 --> 00:03:49.320
I'm not going to try to prove
this thing analytically.
00:03:49.320 --> 00:03:53.740
But as I've said before, the
analytic proof virtually is
00:03:53.740 --> 00:03:56.990
just an abstraction of what
we're doing over here.
00:03:56.990 --> 00:03:59.650
Let's take a look and see
what happens over here.
00:03:59.650 --> 00:04:03.105
Notice that our first
term is 1.
00:04:03.105 --> 00:04:06.005
Our next term is 1/2.
00:04:06.005 --> 00:04:10.670
Our next term is 1 minus 1/2
plus 1/3, which is 5/6.
00:04:10.670 --> 00:04:14.730
What I'm driving at is that if
you compute the sums, 1 minus
00:04:14.730 --> 00:04:17.970
1/2 plus 1/3 minus 1/4, et
cetera, in the order in which
00:04:17.970 --> 00:04:20.630
they're given, what you find
is that the sequence of
00:04:20.630 --> 00:04:30.040
partial sums is 1, 1/2, 5/6,
7/12, 47/60, 13/20.
00:04:30.040 --> 00:04:32.660
Now the thing that I want you to
see, the reason I'm waving
00:04:32.660 --> 00:04:35.700
my hand here and why I think
this helps in the lecture is
00:04:35.700 --> 00:04:36.870
look what's happening.
00:04:36.870 --> 00:04:40.190
You see because the terms
alternate in sign what this
00:04:40.190 --> 00:04:46.640
means is that as I start with 's
sub 1' over here, the next
00:04:46.640 --> 00:04:49.600
term will be to the
left of 's sub 1'.
00:04:49.600 --> 00:04:52.120
Then the term after that
will be to the right.
00:04:52.120 --> 00:04:53.440
Then to the left.
00:04:53.440 --> 00:04:54.550
Then to the right.
00:04:54.550 --> 00:04:55.590
Then to the left.
00:04:55.590 --> 00:04:56.460
Then to the right.
00:04:56.460 --> 00:04:58.230
They keep alternating
this way.
00:04:58.230 --> 00:05:03.500
Moreover, since the terms
decrease in magnitude, it
00:05:03.500 --> 00:05:06.750
means that each jump is less
than the jump that came
00:05:06.750 --> 00:05:08.060
immediately before.
00:05:08.060 --> 00:05:10.910
In other words, as I jumped from
here to here, when I jump
00:05:10.910 --> 00:05:13.310
back I don't come back
quite as far.
00:05:13.310 --> 00:05:16.060
In other words, what I'm doing
now is I'm closing in.
00:05:16.060 --> 00:05:20.140
You see the odd subscripts and
the even subscripts are sort
00:05:20.140 --> 00:05:21.400
of segregated.
00:05:21.400 --> 00:05:22.960
You see what's happening
over here?
00:05:22.960 --> 00:05:28.590
And finally, because the limit
of the n-th term is 0, it
00:05:28.590 --> 00:05:30.370
means that this spacing--
00:05:30.370 --> 00:05:32.420
see the difference between
successive partial
00:05:32.420 --> 00:05:33.920
sums is the n-th term.
00:05:33.920 --> 00:05:35.380
That must go to 0.
00:05:35.380 --> 00:05:38.000
In other words, our limit
'L' is in here.
00:05:38.000 --> 00:05:42.120
And as 'n' increases, the
squeeze is put on and we get
00:05:42.120 --> 00:05:43.280
the existence of a limit.
00:05:43.280 --> 00:05:45.720
For example, whatever
'L' is, it must be
00:05:45.720 --> 00:05:49.500
between 13/20 and 47/60.
00:05:49.500 --> 00:05:51.910
By the way, I say this in
the form of an aside.
00:05:51.910 --> 00:05:53.270
It turns out--
00:05:53.270 --> 00:05:56.130
and for those of us who haven't
seen this before, it's
00:05:56.130 --> 00:05:57.330
a very mystic result.
00:05:57.330 --> 00:05:58.990
That's why I say, who'd
have guessed it?
00:05:58.990 --> 00:06:02.420
It turns out that 'L' is
actually the natural log of 2.
00:06:02.420 --> 00:06:05.380
And the reason I point this
out is that again, you may
00:06:05.380 --> 00:06:08.090
recall that in our notes
we talked about 'Cauchy
00:06:08.090 --> 00:06:11.540
convergence', meaning what do
you do when you don't know how
00:06:11.540 --> 00:06:12.740
to guess the limit?
00:06:12.740 --> 00:06:16.180
You see, the idea here is notice
that this particular
00:06:16.180 --> 00:06:18.250
series converges.
00:06:18.250 --> 00:06:20.820
But we don't know what the limit
is other than the fact
00:06:20.820 --> 00:06:22.890
that it's being squeezed
in over here.
00:06:22.890 --> 00:06:25.840
You see here's a case where we
know that a limit exists, but
00:06:25.840 --> 00:06:28.740
it's particularly difficult
to explicitly name
00:06:28.740 --> 00:06:30.010
what that limit is.
00:06:30.010 --> 00:06:34.120
But that fact notwithstanding,
is it clear that because of
00:06:34.120 --> 00:06:36.930
the fact that the terms
alternate, the magnitudes
00:06:36.930 --> 00:06:40.530
decrease, and the limit is 0
that these things do converge
00:06:40.530 --> 00:06:41.960
to a limit?
00:06:41.960 --> 00:06:43.350
I think it is clear.
00:06:43.350 --> 00:06:46.150
But the next question, and I
apologize for what looks like
00:06:46.150 --> 00:06:49.440
slang here, but I think this is
exactly what's going on in
00:06:49.440 --> 00:06:50.460
your minds right now.
00:06:50.460 --> 00:06:52.620
So what?
00:06:52.620 --> 00:06:55.880
What does this have to do with
what came before and what will
00:06:55.880 --> 00:06:56.780
come later?
00:06:56.780 --> 00:07:00.480
And we're going to see again,
a very, very strange thing
00:07:00.480 --> 00:07:04.120
that happens with infinite sums
that does not happen with
00:07:04.120 --> 00:07:05.310
finite sums.
00:07:05.310 --> 00:07:08.930
Let me lead into that
fairly gradually.
00:07:08.930 --> 00:07:12.800
First of all, I claim that
this particular series--
00:07:12.800 --> 00:07:14.740
see, again, don't get
misled by this.
00:07:14.740 --> 00:07:16.230
It's just a fancy way
of saying what?
00:07:16.230 --> 00:07:21.230
1 minus 1/2 plus 1/3, et
cetera, converges.
00:07:21.230 --> 00:07:24.770
But because the plus terms
cancel the minus terms, the
00:07:24.770 --> 00:07:29.740
pluses cancel the minuses, not
because the terms get small
00:07:29.740 --> 00:07:30.660
fast enough.
00:07:30.660 --> 00:07:32.670
What I mean by that is this.
00:07:32.670 --> 00:07:34.390
Forget about the
signs in here.
00:07:34.390 --> 00:07:36.790
Replace each term by
its magnitude.
00:07:36.790 --> 00:07:38.990
And by the way, that's where
the name 'absolute'
00:07:38.990 --> 00:07:40.980
convergence is going
to come from.
00:07:40.980 --> 00:07:45.060
Namely, the magnitude of a term
is its absolute value.
00:07:45.060 --> 00:07:46.640
And this is what we're going to
be talking about, but the
00:07:46.640 --> 00:07:47.520
idea is this.
00:07:47.520 --> 00:07:50.760
If we replace each term
by its magnitude,
00:07:50.760 --> 00:07:52.500
we obtain the series.
00:07:52.500 --> 00:07:55.810
Summation 'n' goes from 1
to infinity '1 over n'.
00:07:55.810 --> 00:08:00.450
And in the last assignment, we
saw in the exercises that this
00:08:00.450 --> 00:08:03.000
diverged by the integral test.
00:08:03.000 --> 00:08:06.400
In other words, if we leave
out the signs, the series
00:08:06.400 --> 00:08:09.060
diverges because evidently these
terms don't get small
00:08:09.060 --> 00:08:09.575
fast enough.
00:08:09.575 --> 00:08:10.750
Okay.
00:08:10.750 --> 00:08:13.160
Let me state a definition.
00:08:13.160 --> 00:08:14.620
The definition is simply this.
00:08:17.140 --> 00:08:20.720
The series, summation 'n' goes
from 1 to infinity, 'a sub n'
00:08:20.720 --> 00:08:28.300
is said to converge absolutely
if the series that you get by
00:08:28.300 --> 00:08:33.179
replacing each term by its
magnitude converges.
00:08:33.179 --> 00:08:36.159
Now I leave for the
supplementary notes the proof
00:08:36.159 --> 00:08:40.130
that if a series converges
absolutely, it converges in
00:08:40.130 --> 00:08:40.750
the first place.
00:08:40.750 --> 00:08:42.970
In other words, I think it's
rather clear if you look at
00:08:42.970 --> 00:08:45.950
this thing intuitively that if
I replace each term by its
00:08:45.950 --> 00:08:48.660
magnitude, I'll disregard the
plus and minus signs.
00:08:48.660 --> 00:08:51.780
And that resulting series
converges, then the original
00:08:51.780 --> 00:08:54.910
series must've converged also
because the terms couldn't be
00:08:54.910 --> 00:08:56.380
any bigger than this.
00:08:56.380 --> 00:08:59.600
By the way, the formal proof is
kind of messy in places and
00:08:59.600 --> 00:09:01.570
so this is why, as I say,
I leave this for the
00:09:01.570 --> 00:09:03.860
supplementary notes.
00:09:03.860 --> 00:09:07.040
But at any rate that's what we
mean by absolute convergence.
00:09:07.040 --> 00:09:09.380
If you're given a series,
you replace each
00:09:09.380 --> 00:09:11.190
term by it's magnitude.
00:09:11.190 --> 00:09:15.250
If that series converges, we
call the original series
00:09:15.250 --> 00:09:16.830
absolutely convergent.
00:09:16.830 --> 00:09:21.160
Notice by the way, the tie
in now between absolute
00:09:21.160 --> 00:09:23.120
convergence and positive
series.
00:09:23.120 --> 00:09:26.610
Namely, by definition, the
absolute value of 'a sub n' is
00:09:26.610 --> 00:09:27.940
at least as big as 0.
00:09:27.940 --> 00:09:30.420
Consequently, when we're
testing for absolute
00:09:30.420 --> 00:09:33.880
convergence, the series that
we test is positive.
00:09:33.880 --> 00:09:36.620
And we have tests
for convergence
00:09:36.620 --> 00:09:38.460
for positive series.
00:09:38.460 --> 00:09:41.410
Now the sequel to definition
one is of course definition
00:09:41.410 --> 00:09:42.990
two, and that says what?
00:09:42.990 --> 00:09:47.040
A series which converges but
not absolutely is called
00:09:47.040 --> 00:09:48.610
conditionally convergent.
00:09:48.610 --> 00:09:52.540
In other words, it converges on
the condition that the sine
00:09:52.540 --> 00:09:54.840
stay exactly the way they are.
00:09:54.840 --> 00:09:58.340
An example of a conditionally
convergent series is the one
00:09:58.340 --> 00:10:00.020
that we're dealing
with right now.
00:10:00.020 --> 00:10:02.680
Namely, with the pluses and
minuses in there, we just
00:10:02.680 --> 00:10:04.440
showed that the series
converges.
00:10:04.440 --> 00:10:08.000
However, if we replace each
term by its magnitude, the
00:10:08.000 --> 00:10:10.990
resulting series is summation
'1 over n'.
00:10:10.990 --> 00:10:14.230
And that as we saw, diverges.
00:10:14.230 --> 00:10:17.740
Now the question is, what's
so bad about conditional
00:10:17.740 --> 00:10:18.840
convergence?
00:10:18.840 --> 00:10:21.490
What difference does it make
whether a series converges
00:10:21.490 --> 00:10:23.530
absolutely or conditionally?
00:10:23.530 --> 00:10:25.790
Is there any problem
that comes up?
00:10:25.790 --> 00:10:29.370
As I said before, a fantastic
subtlety that occurs, a
00:10:29.370 --> 00:10:32.610
subtlety that has no parallel
in our knowledge of finite
00:10:32.610 --> 00:10:33.710
arithmetic.
00:10:33.710 --> 00:10:35.650
The subtlety is this.
00:10:35.650 --> 00:10:37.470
In fact, I call it that, the
subtlety of conditional
00:10:37.470 --> 00:10:38.500
convergence.
00:10:38.500 --> 00:10:40.570
Namely, the sum of a
00:10:40.570 --> 00:10:42.810
conditionally convergent series--
00:10:42.810 --> 00:10:45.200
and this fantastic--
00:10:45.200 --> 00:10:49.440
depends on the order in which
you write the terms.
00:10:49.440 --> 00:10:53.870
In other words, if the series
converges, but conditionally
00:10:53.870 --> 00:10:57.680
if you change the order of the
terms, surprising as it may
00:10:57.680 --> 00:11:00.970
seem, you actually
change the sum.
00:11:00.970 --> 00:11:03.820
And the best way to do this
I think at this stage, a
00:11:03.820 --> 00:11:07.060
generalization is given in
the supplementary notes.
00:11:07.060 --> 00:11:09.800
But the idea is, let me just
do this in terms of the
00:11:09.800 --> 00:11:11.380
problem that we were
dealing with.
00:11:11.380 --> 00:11:13.820
Let's take the terms of our
conditionally convergent
00:11:13.820 --> 00:11:17.570
series, 1 minus 1/2 plus 1/3,
et cetera, and divide them
00:11:17.570 --> 00:11:19.180
into two teams.
00:11:19.180 --> 00:11:22.700
And if this expression bothers
you, call the teams sets and
00:11:22.700 --> 00:11:25.000
that makes it much more
mathematical.
00:11:25.000 --> 00:11:27.880
Let the first set
consist of the
00:11:27.880 --> 00:11:29.310
positive terms of a series.
00:11:29.310 --> 00:11:32.320
Namely, 1, 1/3, 1/5.
00:11:32.320 --> 00:11:36.450
And in general, '1 over '2n
minus 1'' where 'n' is any
00:11:36.450 --> 00:11:38.500
positive whole number.
00:11:38.500 --> 00:11:42.030
The negative numbers of our
team are minus 1/2--
00:11:42.030 --> 00:11:43.950
or the set 'N', which I'll call
the negative members is
00:11:43.950 --> 00:11:46.550
minus 1/2, minus
1/4, minus 1/6.
00:11:46.550 --> 00:11:49.060
In general, minus '1 over 2n'.
00:11:49.060 --> 00:11:52.980
Now again, by the integral
test, we saw in our last
00:11:52.980 --> 00:11:57.700
lesson that both these series
summation '1 all over '2n
00:11:57.700 --> 00:12:02.500
minus 1'' and summation '1 over
2n' diverge to infinity
00:12:02.500 --> 00:12:04.430
by the integral test.
00:12:04.430 --> 00:12:07.140
Now what my claim is, is that
because both of these two
00:12:07.140 --> 00:12:11.870
series diverge, I can now
rearrange my terms to get any
00:12:11.870 --> 00:12:13.150
sum that I want.
00:12:13.150 --> 00:12:16.540
Well, for example, suppose
somebody says to me, make the
00:12:16.540 --> 00:12:19.960
sum come out to be 3/2.
00:12:19.960 --> 00:12:22.170
As I'll mention later, there's
nothing sacred about 3/2.
00:12:22.170 --> 00:12:24.680
In fact, I'm mentioning that
now but I'll say it again
00:12:24.680 --> 00:12:25.860
later for emphasis.
00:12:25.860 --> 00:12:29.300
I just wanted to pick a number
that wouldn't be too unwieldy.
00:12:29.300 --> 00:12:30.610
But here's what the gist is.
00:12:30.610 --> 00:12:32.850
I want to make the
sum at least 3/2.
00:12:32.850 --> 00:12:36.050
So what I do is I start with my
positive terms from the set
00:12:36.050 --> 00:12:37.500
'P' and add them up.
00:12:37.500 --> 00:12:41.360
1 plus 1/3 plus 1/5,
et cetera.
00:12:41.360 --> 00:12:44.700
The thing that I'm sure of is
that eventually this sum must
00:12:44.700 --> 00:12:46.120
exceed 3/2.
00:12:46.120 --> 00:12:47.560
Why do I know that?
00:12:47.560 --> 00:12:50.270
Well, 'P' diverges
to infinity.
00:12:50.270 --> 00:12:54.400
How could 'P' possibly diverge
to infinity if the sum could
00:12:54.400 --> 00:12:57.730
never-- the sum of the terms in
'P' diverges to infinity.
00:12:57.730 --> 00:12:59.350
How could that happen if
the sum never got at
00:12:59.350 --> 00:13:01.630
least as big as 3/2?
00:13:01.630 --> 00:13:05.850
So the idea is I write down all
of these terms, add them
00:13:05.850 --> 00:13:09.810
up, until my sum first exceeds
or equals 3/2.
00:13:09.810 --> 00:13:12.740
And as I say, this must
happen because the
00:13:12.740 --> 00:13:14.560
series diverges to infinity.
00:13:14.560 --> 00:13:17.790
Well, in particular, I
observe that 1 plus
00:13:17.790 --> 00:13:21.110
1/3 plus 1/5 is 23/15.
00:13:21.110 --> 00:13:23.970
And that's now, for the first
time, bigger than 3/2.
00:13:23.970 --> 00:13:27.460
What I do next is I annex the
negative members, in other
00:13:27.460 --> 00:13:29.810
words, the members of capital
'N', until the
00:13:29.810 --> 00:13:32.540
sum falls below 3/2.
00:13:32.540 --> 00:13:35.580
Watch what I'm doing here.
00:13:35.580 --> 00:13:36.930
I stopped at 1/5.
00:13:36.930 --> 00:13:39.010
I have 1 plus 1/3 plus 1/5.
00:13:39.010 --> 00:13:40.710
Now I subtract 1/2.
00:13:40.710 --> 00:13:44.710
That gives me 31/30 and
that's less than 3/2.
00:13:44.710 --> 00:13:46.810
So now my sum is below 3/2.
00:13:46.810 --> 00:13:51.250
What I do next is I continue
with 'P' where I left off.
00:13:51.250 --> 00:13:53.130
See I left off with 1/5.
00:13:53.130 --> 00:14:00.420
I now start tacking on 1/7, 1/9,
1/11, 1/13, et cetera.
00:14:00.420 --> 00:14:04.380
Until the sum, again,
exceeds 3/2.
00:14:04.380 --> 00:14:07.050
Now how do I know that the
sum has to exceed 3/2?
00:14:07.050 --> 00:14:11.520
Well, remember when we add up
all of the members of 'P', we
00:14:11.520 --> 00:14:13.170
get a divergent series.
00:14:13.170 --> 00:14:16.980
That means the sum increases
without bound.
00:14:16.980 --> 00:14:20.120
As we've mentioned many times so
far in our course, that if
00:14:20.120 --> 00:14:24.430
you chop off a finite number
of terms from a divergent
00:14:24.430 --> 00:14:26.830
series, the remaining
series, what's
00:14:26.830 --> 00:14:28.920
left, still must diverge.
00:14:28.920 --> 00:14:34.550
In other words, if this series
1 plus 1/3 plus 1/5 plus 1/7
00:14:34.550 --> 00:14:38.190
diverges to infinity, the fact
that I chop off those terms
00:14:38.190 --> 00:14:41.740
that add up to just an excess
of 3/2, what's left is still
00:14:41.740 --> 00:14:43.380
going to diverge to infinity.
00:14:43.380 --> 00:14:45.520
So I can keep on
going this way.
00:14:45.520 --> 00:14:47.450
What I do is I add on 1/7.
00:14:47.450 --> 00:14:50.830
The result turns out to
be 247/210, which is
00:14:50.830 --> 00:14:52.320
still less than 3/2.
00:14:52.320 --> 00:14:54.080
And to spare you the
gory details--
00:14:54.080 --> 00:14:55.690
and believe me, they are gory.
00:14:55.690 --> 00:14:57.320
I worked it out myself.
00:14:57.320 --> 00:15:00.030
Without a desk calculator
this gets to be a mess.
00:15:00.030 --> 00:15:03.480
It turns out that when I add on
1/13, get down to here, the
00:15:03.480 --> 00:15:06.570
sum is this, which is
still less than 3/2.
00:15:06.570 --> 00:15:08.600
But then I add on 1/15.
00:15:08.600 --> 00:15:11.640
The sum gets to be this, which
I simply call 'k'.
00:15:11.640 --> 00:15:14.360
That turns out to be
greater than 3/2.
00:15:14.360 --> 00:15:17.330
Then you see what I do is
return to my series of
00:15:17.330 --> 00:15:21.140
negative terms, tack those on
till the sum falls below.
00:15:21.140 --> 00:15:24.970
And what's happening here
pictorially is the following.
00:15:24.970 --> 00:15:27.660
You see what happened
was 3/2--
00:15:27.660 --> 00:15:31.290
I added on terms till
I exceeded 3/2.
00:15:31.290 --> 00:15:33.730
That was 23/15.
00:15:33.730 --> 00:15:37.580
Then I get down below 3/2.
00:15:37.580 --> 00:15:39.760
Then up again above 3/2.
00:15:39.760 --> 00:15:41.860
And still sparing
you the details,
00:15:41.860 --> 00:15:43.220
notice what's happening.
00:15:43.220 --> 00:15:47.280
Each time that I passed 3/2, I
pass it by less than before
00:15:47.280 --> 00:15:49.500
because the terms are getting
smaller in magnitude.
00:15:49.500 --> 00:15:54.080
What's happening is and I hope
this crazy little diagram here
00:15:54.080 --> 00:15:55.110
serves the purpose.
00:15:55.110 --> 00:15:59.910
You see what's happening is
I'm zeroing in on 3/2.
00:15:59.910 --> 00:16:03.690
In other words, this particular
rearrangement will
00:16:03.690 --> 00:16:07.230
guarantee me that those terms
will add up to 3/2.
00:16:07.230 --> 00:16:11.050
Now again, as I said before, 3/2
was not important, though
00:16:11.050 --> 00:16:12.430
the arithmetic gets messier.
00:16:12.430 --> 00:16:14.960
And again, that's the best
word I can think of.
00:16:14.960 --> 00:16:17.150
In other words, I went through
several sheets of paper just
00:16:17.150 --> 00:16:19.650
trying to get to the next
stage over here before I
00:16:19.650 --> 00:16:21.250
realized it wasn't worth it.
00:16:21.250 --> 00:16:23.280
I mean it's something we
can all do on our own.
00:16:23.280 --> 00:16:26.490
But the larger number that you
choose, the more terms you're
00:16:26.490 --> 00:16:29.420
going to have to add up before
you exceed this.
00:16:29.420 --> 00:16:31.110
Don't confuse two things here.
00:16:31.110 --> 00:16:34.900
I obviously have to add up an
awful lot of terms of the form
00:16:34.900 --> 00:16:40.680
1, 1/3, 1/5, 1/7, 1/9, 1/11
to get, say a million.
00:16:40.680 --> 00:16:43.610
But the point is that since
that series diverges,
00:16:43.610 --> 00:16:45.820
eventually by going
out far enough--
00:16:45.820 --> 00:16:48.180
now far enough might be
billions of terms.
00:16:48.180 --> 00:16:52.120
But still a finite number, the
sum will exceed 1 million.
00:16:52.120 --> 00:16:53.710
That's the key point.
00:16:53.710 --> 00:16:56.040
In other words, I can keep
oscillating around any sum
00:16:56.040 --> 00:16:59.040
that I want just by exceeding
it, coming back with a
00:16:59.040 --> 00:17:01.500
negative terms, getting less
than that, and alternating
00:17:01.500 --> 00:17:02.660
back and forth.
00:17:02.660 --> 00:17:06.329
Again, this will be left for
much greater detail for the
00:17:06.329 --> 00:17:09.329
supplementary notes
and the exercises.
00:17:09.329 --> 00:17:11.730
I'll mention a little bit more
about that in a few minutes.
00:17:11.730 --> 00:17:14.680
But the summary so
far is this.
00:17:14.680 --> 00:17:17.970
If the series summation 'n' goes
from 1 to infinity, 'a
00:17:17.970 --> 00:17:22.200
sub n' is conditionally
convergent, its limit exists.
00:17:22.200 --> 00:17:23.869
Let's not forget that.
00:17:23.869 --> 00:17:25.260
Its limit exists.
00:17:25.260 --> 00:17:28.850
But that limit depends on not
changing the order in which
00:17:28.850 --> 00:17:29.820
the terms were given.
00:17:29.820 --> 00:17:32.940
In other words, the limit
changes as the order of the
00:17:32.940 --> 00:17:34.690
terms is changed.
00:17:34.690 --> 00:17:39.420
That is, rearranging the terms
actually changes the series.
00:17:39.420 --> 00:17:42.370
And there is nothing in finite
arithmetic that is
00:17:42.370 --> 00:17:43.220
comparable to this.
00:17:43.220 --> 00:17:46.260
In other words, if you have 50
numbers to add up, or 50
00:17:46.260 --> 00:17:49.150
million numbers, or 50 billion
numbers, no matter how you
00:17:49.150 --> 00:17:54.880
rearrange those numbers, the
sum exists and is the same
00:17:54.880 --> 00:17:58.660
independently of what the
rearrangement is.
00:17:58.660 --> 00:18:01.500
This is not comparable
to finite arithmetic.
00:18:01.500 --> 00:18:03.940
And the moral is-- and again, I
say this in slang expression
00:18:03.940 --> 00:18:05.170
because I want this
to rub off.
00:18:05.170 --> 00:18:06.620
I want you to remember this.
00:18:06.620 --> 00:18:09.620
Don't monkey with conditional
convergence.
00:18:09.620 --> 00:18:13.790
If the series is conditionally
convergent, make sure that you
00:18:13.790 --> 00:18:17.366
add the terms in the order
in which they appear.
00:18:17.366 --> 00:18:20.360
That if you change the
order, you will
00:18:20.360 --> 00:18:21.970
get a different limit.
00:18:21.970 --> 00:18:24.660
And what will happen is you'll
get a limit that is the right
00:18:24.660 --> 00:18:26.990
answer to the wrong problem.
00:18:26.990 --> 00:18:28.450
In other words, changing
the order of the
00:18:28.450 --> 00:18:30.870
terms changes the limit.
00:18:30.870 --> 00:18:33.370
And this is why conditional
convergence is
00:18:33.370 --> 00:18:34.750
particularly annoying.
00:18:34.750 --> 00:18:36.990
It means that all of these
things that come natural an
00:18:36.990 --> 00:18:39.970
ordinary arithmetic are
lacking in conditional
00:18:39.970 --> 00:18:41.540
convergence.
00:18:41.540 --> 00:18:44.160
Now, what does the
sequel to this?
00:18:44.160 --> 00:18:48.110
The sequel is that all is well
when you have absolute
00:18:48.110 --> 00:18:49.840
convergence.
00:18:49.840 --> 00:18:51.680
I just wrote this out to make
sure that we have this in
00:18:51.680 --> 00:18:52.520
front of us.
00:18:52.520 --> 00:18:56.400
The beauty of absolute
convergence is that the sum of
00:18:56.400 --> 00:18:59.890
an absolutely convergent series
is the same for every
00:18:59.890 --> 00:19:01.770
rearrangement of the terms.
00:19:01.770 --> 00:19:05.060
The details are left to the
supplementary notes.
00:19:05.060 --> 00:19:09.110
Now, what am I trying to bring
out by all of this?
00:19:09.110 --> 00:19:14.270
You see, the beauty of positive
series is that every
00:19:14.270 --> 00:19:17.750
time we talk about absolute
convergence, the test involves
00:19:17.750 --> 00:19:18.650
a positive series.
00:19:18.650 --> 00:19:21.540
In other words, by knowing how
to test positive series for
00:19:21.540 --> 00:19:24.830
convergence, we can test
any series for absolute
00:19:24.830 --> 00:19:25.800
convergence.
00:19:25.800 --> 00:19:28.270
What is the beauty of absolute
convergence?
00:19:28.270 --> 00:19:31.790
The beauty of absolute
convergence is that we can
00:19:31.790 --> 00:19:34.260
rearrange the terms in any order
that we want if it's
00:19:34.260 --> 00:19:36.570
convenient to pick a different
order than another.
00:19:36.570 --> 00:19:40.030
And the sum will not depend
on this rearrangement.
00:19:40.030 --> 00:19:42.330
You see the point is we are
not saying keep away from
00:19:42.330 --> 00:19:44.160
conditionally convergent
series.
00:19:44.160 --> 00:19:47.930
In many important applications
you have to come to grips with
00:19:47.930 --> 00:19:49.330
conditional convergence.
00:19:49.330 --> 00:19:52.270
All we are saying is that if
you want to be able to fool
00:19:52.270 --> 00:19:55.620
around numerically with these
series, if you don't have
00:19:55.620 --> 00:19:58.590
absolute convergence, you're
in a bit of trouble.
00:19:58.590 --> 00:20:01.500
Now you see, the point is that
our textbook does a very good
00:20:01.500 --> 00:20:05.030
job in talking about absolute
convergence versus conditional
00:20:05.030 --> 00:20:05.840
convergence.
00:20:05.840 --> 00:20:09.180
But for some reason, does not
mention the problem of
00:20:09.180 --> 00:20:10.900
rearranging terms.
00:20:10.900 --> 00:20:13.640
And therefore, much of what I've
talked about today, the
00:20:13.640 --> 00:20:16.920
importance of absolute
convergence, is in terms of
00:20:16.920 --> 00:20:17.990
rearrangements.
00:20:17.990 --> 00:20:20.730
And because this material is
not in the textbook, what I
00:20:20.730 --> 00:20:24.900
have elected to do is to put
all of this material that
00:20:24.900 --> 00:20:28.420
we've talked about today almost
verbatim except in a
00:20:28.420 --> 00:20:31.680
more generalized form, into
the supplementary notes,
00:20:31.680 --> 00:20:34.510
supplying whatever proofs are
necessary and whatever
00:20:34.510 --> 00:20:36.460
intuitive ideas are necessary.
00:20:36.460 --> 00:20:39.350
At any rate, read the
supplementary notes, do the
00:20:39.350 --> 00:20:43.370
exercises, and we'll continue
our discussion next time.
00:20:43.370 --> 00:20:44.700
And until next time, goodbye.
00:20:47.780 --> 00:20:50.310
ANNOUNCER: Funding for the
publication of this video was
00:20:50.310 --> 00:20:55.030
provided by the Gabriella and
Paul Rosenbaum Foundation.
00:20:55.030 --> 00:20:59.200
Help OCW continue to provide
free and open access to MIT
00:20:59.200 --> 00:21:03.400
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