Riemann series rearrangement theorem

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This article describes a test that is used to determine, in some cases, whether a given infinite series or improper integral converges. It may help determine whether we have absolute convergence, conditional convergence, or neither.
View a complete list of convergence tests

Statement

Consider a series:

$\sum _{k=1}^{\infty }a_{k}$

(Note: the starting point of the summation does not matter for the theorem).

Suppose the series is a conditionally convergent series: it is a convergent series but not an absolutely convergent series, i.e., the series $\sum _{k=1}^{\infty }|a_{k}|$ does not converge.

Then, the following are true:

1. $\lim _{k\to \infty }a_{k}=0$
2. The sub-series comprising only those $a_{k}$s that are positive diverges.
3. The sub-series comprising only those $a_{k}$s that are negative diverges.
4. Given any two elements $L\leq U$ in $[-\infty ,\infty ]$ (i.e., they could be real numbers, or $\pm \infty$) there exists a rearrangement of the $a_{k}$s such that the limit inferior of the partial sums is $L$ and the limit superior of the partial sums is $U$. In particular, since we are allowed to set $L=U$, we can obtain a rearrangement that converges to any desired sum.

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Full timed transcript: [SHOW MORE]

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

• Levy-Steinitz theorem is a generalization to series of vectors in $\mathbb {R} ^{n}$. The claim is that the set of possible sums of rearrangements of any series of vectors that are finite vectors, if non-empty, is an affine subspace of $\mathbb {R} ^{n}$.

Proof

Proof of (1), (2), and (3)

(1) is true on account of convergence.

(2) and (3): We can show that if any one of the sums is finite, so is the other one, and both being finite would force absolute convergence.

Proof of (4)

We outline a constructive procedure to create the series. For simplicity, we will assume that none of the $a_{k}$s are 0. The proof can be modified somewhat to include the case of some of the $a_{k}$s being zero.

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Case of finite values of limit superior and limit inferior

Setup (unsorted version):

• Pick the sub-series of all the positive value terms.
• Pick the sub-series of all the negative value terms.

Setup (sorted version)

• Arrange all the positive values among the $a_{k}$s in decreasing order. We will always pick from the list in the order arranged, i.e., we will always pick the largest positive value currently in the list when asked to pick from the list. The fact that we can list all the positive $a_{k}$s in decreasing order relies on $\lim _{k\to \infty }a_{k}=0$.
• Arrange all the negative values among the $a_{k}$s in decreasing order of magnitude (hence, increasing order). We will always pick from the list in the order arranged, i.e., we will always pick the largest magnitude negative value currently in the list when asked to pick from the list. The fact that we can list all the negative $a_{k}$s in increasing order relies on $\lim _{k\to \infty }a_{k}=0$.

Rearrangement creation:

• Begin by picking positive values (using the ordering of terms in the positive sub-series) and keep doing so until the partial sum so far is greater than $U$. Note that we always reach such a point after picking finitely many positive values because the sum of all positive terms of the series is $\infty$.
• Now, start picking negative values (using the ordering of terms in the negative sub-series) till the overall partial sum (including the positive and negative values picked so far) is less than $L$. Again, this is possible because the negative value terms add up to $-\infty$.
• Now, resume picking positive values till the overall partial sum is greater than $U$, then switch back to picking negative values, and so on.

If we are using the sorted version of the setup, we are always picking the largest magnitude term among those left within the terms of that sign. If we are using the unsorted version, we are picking the positive (respectively, negative) terms in the same order as in the original series. However, the nature of alternation between positive and negative terms may be quite far from the original series had.

Checking conditions:

• Every term gets picked exactly once: We note that, because both the positive and negative term sums diverge, it is the case that we cycle back and forth infinitely many times. Each time we cycle, we pick at least one new term from each side, so it's clear that every term gets picked. The picking procedure makes sure every time is picked exactly once.
• The limit superior is $U$: We know that the partial sums become greater than or equal to $U$ infinitely often, so the limit superior is at least $U$. The extent to which the partial sum can overshoot $U$ each time is bounded by the magnitude of the terms, which we have assumed approaches zero. This shows that the limit superior is exactly $U$.
• The limit inferior is $L$: Similar reasoning as with the limit superior.

Case where we allow infinity for one or both the values

• If the limit superior is $\infty$ and the limit inferior is finite: In this case, we construct an increasing sequence of finite numbers $U_{1},U_{2},\dots ,$ that approaches $\infty$ first (we can construct this sequence before looking at the $a_{k}$s). Now, we modify the construction of the rearrangement series as follows: instead of trying to cross $U$ every time with the positive terms, we try to cross $U_{i}$ with the positive terms at the $i^{th}$ stage.
• If the limit superior and limit inferior are both $\infty$: We construct increasing sequences to $\infty$ in place of both $U$ and $L$.
• If the limit inferior is $-\infty$ and the limit superior is finite: In this case, we construct a decreasing sequence of finite numbers $L_{1},L_{2},\dots$ that approaches $-\infty$ first (we can construct this sequence without looking at the $a_{k}$s). Now, we modify the construction of the rearrangement series as follows: instead of trying to cross $L$ every time with the negative terms, we try to cross $L_{i}$ with the negative terms at the $i^{th}$ stage.
• If the limit superior and limit inferior are both $-\infty$: We construct decreasing sequences to $-\infty$ in place of both $U$ and $L$.
• If the limit superior is $\infty$ and the limit inferior is $-\infty$: We construct an increasing sequence to $\infty$ in place of $U$ and a decreasing sequence to $-\infty$ in place of $L$.