0:00:16.670,0:00:20.240
Vipul: Okay, so this talk is going to be about
the Riemann Series Rearrangement Theorem.
0:00:20.240,0:00:24.700
Some people just call it the Riemann Series
Theorem and it's about certain kinds of series.
0:00:24.700,0:00:30.180
Series is something like this: You have summation
k equals 1 to infinity a sub k [symbolically:
0:00:30.180,0:00:34.650
a_k]. So infinite sum, how is the sum defined?
0:00:34.650,0:00:39.230
Rui: The sum is defined as the sum of all
the terms.
0:00:39.230,0:00:43.290
Vipul: Yes, but it is defined as the limit
of something, right, limit of what?
0:00:43.290,0:00:44.809
Rui: I'm not sure.
0:00:44.809,0:00:49.400
Vipul: Well, it's defined as... if you want
to add up infinitely many terms, you cannot
0:00:49.400,0:00:56.400
add them all at once. So you add up the first,
let's write this down. You have a series a_1
0:01:01.510,0:01:08.510
+ a_2 + a_3 + ... there's the nth term. How
would you add this up? How do you find this
0:01:11.750,0:01:15.570
infinite
sum? What would you compute? Well you'd say,
0:01:15.570,0:01:22.570
first we do a_1 then you add a_1 + a_2, then
you do a_1 + a_2 + a_3 right? Then take the
0:01:34.939,0:01:41.310
limit of these things. So what is it? It's
the limit as n goes to infinity of what?
0:01:41.310,0:01:46.619
Rui: The sum of a_k, k going from 1 to n.
0:01:46.619,0:01:51.369
Vipul: Yes, exactly what I was saying. These
sums are called the... these things whose
0:01:51.369,0:01:57.450
limit you're taking are called the what? They
are called the *partial sums*, okay?
0:01:57.450,0:01:59.170
Rui: Okay.
0:01:59.170,0:02:06.170
Vipul: So, in particular, it matters like
in what order you write them. Like this series
0:02:09.560,0:02:15.280
that you're taking a_1 then a_1 + a_2 then
a_1 + a_2 + a_3 and so on and we'll see why
0:02:15.280,0:02:19.490
that is significant. But this is the former
definition. If this limit exists then that's
0:02:19.490,0:02:24.670
the series sum. If the limit doesn't exist
then the series cannot be summed. By the way,
0:02:24.670,0:02:31.670
if you just have a series and I say the sum
exists, than you say that the series *converges*.
0:02:32.180,0:02:35.400
That's terminology which we are hopefully
familiar with.
0:02:35.400,0:02:38.629
Now the series is called *conditionally convergent*
0:02:38.629,0:02:45.629
if it converges but another series which is just
the absolute values of the terms does not
0:02:47.340,0:02:54.340
converge. If the absolute values series converged,
it would be called *absolutely convergent*.
0:02:54.930,0:03:00.569
Conditionally convergent means convergent
but not absolutely convergent. Okay?
0:03:00.569,0:03:07.569
Let me just write down an example, I won't
explain fully why that's so [i.e., I'll skip
0:03:11.140,0:03:15.409
the details] because that may be a little
difficult for some people to understand but
0:03:15.409,0:03:22.409
here is one example of the series that is
conditionally convergent but not absolutely
0:03:22.980,0:03:26.329
convergent. I mean it is convergent but not
absolutely convergent therefore it's
0:03:26.329,0:03:27.079
conditionally convergent.
0:03:27.079,0:03:28.090
[Example series 1 - (1/2) + (1/3) - ...]
0:03:28.090,0:03:31.980
This series is convergent by a result called
the alternating
0:03:31.980,0:03:38.769
series theorem which we have a separate video
on. Basically, the terms are going to zero,
0:03:38.769,0:03:44.540
decreasing in magnitude, and alternating in
sign. If that happens, the series converges,
0:03:44.540,0:03:49.739
okay? It is not absolutely convergent. Why?
Well, what are the absolute values of the
0:03:49.739,0:03:52.129
terms?
What's the series of absolute values of the
0:03:52.129,0:03:52.819
terms?
0:03:52.819,0:03:57.680
Rui: Change all negative signs to positive.
0:03:57.680,0:04:04.680
Vipul: So this series does not converge. You
can see it in many ways. You can see it using
0:04:05.969,0:04:11.090
the integral test; the corresponding integral
does not converge. If you are already familiar
0:04:11.090,0:04:14.760
with the degree difference test, which basically
again follows from the integral test, this
0:04:14.760,0:04:21.760
is like summation of this rational function
and this rational function summation 1 over
0:04:22.570,0:04:27.750
k, here the degree difference
is 1 and if the degree difference is 1 the
0:04:27.750,0:04:33.380
absolute value summation does not converge.
You do have examples of series that are convergent
0:04:33.380,0:04:38.870
but not absolutely convergent. This definition
does get satisfied at least for something.
0:04:38.870,0:04:44.020
Can you tell me what this converges to? The
information I have given you doesn't tell
0:04:44.020,0:04:51.020
you. Do you happen to know what this converges
to? No? Well, it converges to natural log
0:04:51.280,0:04:56.940
of 2 [ln 2 ~ 0.7]. That's not obvious at all.
It follows from some stuff with power series
0:04:56.940,0:05:01.750
which you might see at a later stage. But
it's not important what it converges to. Point
0:05:01.750,0:05:07.430
is it's conditionally convergent. So, here's
the theorem. Actually, it is part 4 that's
0:05:07.430,0:05:14.430
the real theorem, part 1, 2, 3, you can think
of as preliminary things for the theorem.
0:05:14.880,0:05:19.200
Part 1 says that the terms have to go to zero.
That actually follows from it converging.
0:05:19.200,0:05:23.970
If a series converges, the terms have to go
to zero. Do
0:05:23.970,0:05:30.970
you see why? Well, if the sum is some finite
real number, right, here's a series, and the
0:05:32.570,0:05:39.570
sum of the series is L, then the partial sums...
remember, L is the limit of what? Limit as
0:05:46.430,0:05:48.580
n approaches to infinity of what?
0:05:48.580,0:05:50.330
Rui: Partial sum?
0:05:50.330,0:05:57.330
Vipul: Yes. [sum of k^{th} terms], k equals
1 to n, okay? That's good. Now, suppose the
0:06:00.100,0:06:04.740
limit is L which means that eventually, all
the partial sums will be trapped in a small
0:06:04.740,0:06:11.740
neighborhood of L. Right? So if this neighborhood
is of radius epsilon, then all the partial
0:06:11.960,0:06:18.310
sums are within here. How big can the terms
be? What's the maximum size any term can have?
0:06:18.310,0:06:24.220
Like eventually, all the terms will have size
at most, what?
0:06:24.220,0:06:25.340
Rui: epsilon.
0:06:25.340,0:06:29.590
Vipul: Not epsilon. It could go from here
to here and from here to here.
0:06:29.590,0:06:30.450
Rui: Zero.
0:06:30.450,0:06:37.450
Vipul: Well, zero when you take epsilon approaching
zero. But right now, if all the partial sums
0:06:41.250,0:06:45.960
are here in this ball, then what can you say?
The difference between any two things in this
0:06:45.960,0:06:47.340
ball is at most what?
0:06:47.340,0:06:48.680
Rui: Two epsilon [i.e., twice epsilon].
0:06:48.680,0:06:53.990
Vipul: Two epsilon. And any term is the difference
between one partial sum and the next, right?
0:06:53.990,0:07:00.990
So a_1 + a_2 + ... + a_{n-1} and the next
partial sum is a_1 + a_2 + ... + a_{n-1} + a_n.
0:07:02.810,0:07:09.810
You've added a_n,right? If this partial
sum is in the ball, in this interval, and
0:07:10.560,0:07:14.410
if this partial sum is alos in the interval,
then that means the difference a_n has to
0:07:14.410,0:07:18.870
have size less than 2 epsilon. Eventually,
all
0:07:18.870,0:07:25.870
the terms become at most 2 epsilon and therefore
as epsilon goes to zero the terms have to
0:07:26.380,0:07:30.600
go to zero. That's the rough idea and that
doesn't require conditional convergence. That's
0:07:30.600,0:07:33.810
just a fact about convergent series.
0:07:33.810,0:07:36.990
The next two things that are interesting,
it says that if you just look at the positive
0:07:36.990,0:07:43.990
terms, then that sub-series diverges. If you
just look at the negative terms then that
0:07:44.250,0:07:48.000
subseries diverges. Which means the positive
terms add up to infinity and the negative
0:07:48.000,0:07:50.080
terms add up to?
0:07:50.080,0:07:52.340
Rui: Negative infinity.
0:07:52.340,0:07:59.340
Vipul: Negative infinity. Why should that
be true? Suppose the positive terms actually
0:08:01.880,0:08:08.880
added up to something [finite] like... and
here's the series a_1 + a_2 and let's say
0:08:11.400,0:08:18.400
the sum is 4, okay. Suppose the positive terms
add [up] to 13, okay? Now if the positive
0:08:25.730,0:08:29.400
term added up to something finite, the negative
terms would also add up to something finite.
0:08:29.400,0:08:31.440
What should the negative terms add up to?
0:08:31.440,0:08:32.560
Rui: Nine.
0:08:32.560,0:08:33.680
Vipul: Negative.
0:08:33.680,0:08:35.370
Rui: Negative nine.
0:08:35.370,0:08:39.950
Vipul: Now, what should the absolute values
add up to then?
0:08:39.950,0:08:41.500
Rui: Two?
0:08:41.500,0:08:45.640
Vipul: No, the absolute value series, what
would that add up to?
0:08:45.640,0:08:48.380
Rui: Twenty one, twenty two.
0:08:48.380,0:08:52.769
Vipul: Why did you say twenty one first?
0:08:52.769,0:08:55.459
Rui: I have no idea.
0:08:55.459,0:08:59.879
Vipul: Okay. Twenty two, right? What I'm basically
saying is if the positive terms converge and
0:08:59.879,0:09:04.220
the negative terms are also forced to converge,
then the sum of the absolute thing of these
0:09:04.220,0:09:07.269
will be the epsilon and that will converge
and that contradicts our assumption that it's
0:09:07.269,0:09:10.300
not absolutely convergent. Similarly, if the
negative terms
0:09:10.300,0:09:14.189
converged, the positive terms also converge
and then the absolute value will also have
0:09:14.189,0:09:17.759
to converge. Therefore, neither the positive
nor the negative things can converge. The
0:09:17.759,0:09:21.670
positive ones have to diverge and the negative
ones have to diverge. Okay. That's not the
0:09:21.670,0:09:25.220
full formal proof. Just the idea. We will
have to prove various things to
0:09:25.220,0:09:26.120
establish it formally.
0:09:26.120,0:09:31.250
So we're here so far: the terms go to zero,
the positive terms subseries diverges, the
0:09:31.250,0:09:34.199
negative terms subseries diverges, okay?
0:09:34.199,0:09:35.829
Rui: Okay.
0:09:35.829,0:09:42.389
Vipul: Now we come to a really remarkable
fact which is this. Suppose I pick two numbers
0:09:42.389,0:09:46.209
where they're not only numbers, they're allowed
to be minus infinity and infinity. What does
0:09:46.209,0:09:49.740
this notation [referring to [-infinity,infinity]]
mean? It's like all reals, but I'm including
0:09:49.740,0:09:52.579
minus infinity and infinity, okay?
0:09:52.579,0:09:53.550
Rui: Okay.
0:09:53.550,0:09:59.430
Vipul: Suppose I take two things in here.
Again, this one is less than equal to other
0:09:59.430,0:10:02.639
and you know how you compare minus infinity
with ordinary numbers, with each other and
0:10:02.639,0:10:07.740
with infinity. You have two things and they
could be equal but L is less than equal to
0:10:07.740,0:10:12.529
U. So L is
lower and U is upper. Then, there's a rearrangement
0:10:12.529,0:10:17.810
of the a_k's, so you can rearrange, you can
permute the a_k such that with this rearranged
0:10:17.810,0:10:23.249
series, the partial sums have lim inf equals
L and lim sup equals U. So you're wondering
0:10:23.249,0:10:28.480
in lim inf and lim sup are, right?
0:10:28.480,0:10:35.480
Basically, here's your series, summation of,
let's call it b_k now. The partial sum, let's
0:10:54.100,0:11:01.100
define S_n is summation k=1 to
n of b_k. Ordinarily, when you just take what
is the infinite sum, you just take limit and
0:11:10.290,0:11:17.290
approach it to infinity S_n, this is the sum
of the series, right? Now I could also define
0:11:19.459,0:11:26.459
this thing, lim inf as n approaches infinity
S_n. What this is doing is, for every n, what
0:11:31.040,0:11:34.790
it
really is, it is limit as n approaches infinity
0:11:34.790,0:11:41.790
inf of m >= n of
S_m. For every n, it's looking at the glb
0:11:48.589,0:11:55.589
of sums, all the partial sums beyond that.
Then, it's making n approach infinity. What
0:12:00.259,0:12:05.540
that essentially is doing is, imagine this
that you are here, the corresponding list
0:12:05.540,0:12:10.779
of partial sums is: you start with zero then
you add b_1 [said *a_1* incorrectly] then
0:12:10.779,0:12:15.059
you add b_2 [said *a_2* incorrectly] then
you add b_3 [said *a_3* incorrectly] which
0:12:15.059,0:12:16.689
maybe
negative, some of them could be negative,
0:12:16.689,0:12:21.199
some of them could be positive. Then you add
b_4 [said *a_4* incorrectly] so you keep on
0:12:21.199,0:12:28.199
hopping along the number line right? These
points are the partial sums.
0:12:29.499,0:12:36.499
The lim inf of these is sort of saying...
Suppose these partial sums didn't converge?
0:12:38.410,0:12:45.410
Suppose you had a situation where, no they're
sort of going like... They're keeping on oscillating
0:12:47.050,0:12:50.540
between two numbers like
that . Then you want to see the lim inf for
0:12:50.540,0:12:54.319
the smaller number and the lim sup which I
will define later, is the bigger one. The
0:12:54.319,0:13:01.319
point is the lim inf is sort of saying, it's
the smallest thing which keeps occuring. Or
0:13:03.749,0:13:10.749
near which you keep going. Among the things
which you keep sort of going near, it's the
0:13:16.360,0:13:23.360
smallest one, the left most one. Among the
things which you keep on going near. Formally,
0:13:23.430,0:13:30.430
it's just this. It is the limit as n approaches
infinity. Inf just means... is the shorthand for
0:13:33.170,0:13:40.170
the glb if you want. So it's the limit as
n approaches infinity of glb of all the partial
0:13:42.480,0:13:48.050
sums beyond n. You are taking the smallest
thing which keeps occurring up to infinity.
0:13:48.050,0:13:55.050
And similarly, if you have the lim sup, we'll
define similarly, it should be the limit as
0:13:58.740,0:14:05.740
n approaches infinity supremum [another word
for lub] of m greater than equal to n, of
0:14:07.529,0:14:14.009
S_m. Intuitively, if your summation is such
as that you have these two points and your
0:14:14.009,0:14:21.009
summation is partial sums, they are oscillating
between clustering here and clustering around
0:14:21.839,0:14:24.240
this one. Then here you have your lim inf
and here
0:14:24.240,0:14:26.930
you have your lim sup.
0:14:26.930,0:14:31.410
If your partial sums are just converging to
a single point, then that's the limit and
0:14:31.410,0:14:35.009
that's then equal to both the lim inf and
lim sup. But you could have situations where
0:14:35.009,0:14:42.009
the lim inf and lim sup are not the same.
We want to now show this thing which says
0:14:47.149,0:14:52.680
that any pair of numbers, you can arrange
the series in such a way that the lim inf
0:14:52.680,0:14:59.319
is the lower one and the lim sup is the bigger
one. The remarkable thing it's saying is that
0:14:59.319,0:15:06.319
now here, you have your series, say this series
and I've told you the sum is ln 2. What I
0:15:07.329,0:15:12.699
am saying is that... give me some other real
number?
0:15:12.699,0:15:14.309
Rui: 1, 4
0:15:14.309,0:15:20.490
Vipul: 1/4, there is a way of rearranging
this series... give me two real numbers actually.
0:15:20.490,0:15:22.399
Rui: I said 1 and 4.
0:15:22.399,0:15:28.189
Vipul: One and four? So there's a way of rearranging
this series such that the lim inf of the partial
0:15:28.189,0:15:34.970
sums is 1 and the lim sup of the partial sum
is 4. You could also pick one of the things
0:15:34.970,0:15:40.529
to be infinity and one to be negative infinity.
So you could show that there's a way of rearranging
0:15:40.529,0:15:47.220
this series so that the lim inf of the partial
sums is 5 and the lim sup of the partial sums
0:15:47.220,0:15:50.619
is infinity.
0:15:50.619,0:15:57.619
So how would you do this? How would you prove
this? We can do that in a separate video-
0:15:59.860,0:16:02.329
right? The construction.