Video:Riemann series rearrangement theorem
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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.Related facts
Proof
Proof of (1), (2), and (3)
(see video above)