Perhaps the first significant observation in the theory of infinite series is that there are convergent series whose terms can be rearranged to form a new series that converges to a different value.
A well known example is provided by the alternating harmonic series,
and its rearrangement
Henry Parker Manning. Irrational numbers and their representation by sequences and series. John Wiley & Sons, 1906,
Laurent evaluated the latter by inserting parentheses (see pages 97, 98):
A similar argument is possible with the rearrangement
which can be rewritten as
The first person to realize that rearranging the terms of a series may change its sum was Dirichlet in 1827, while working on the convergence of Fourier series. (The date is mentioned by Riemann in his Habilitationsschrift, see also page 94 of Ivor Grattan-Guinness. The Development of the Foundations of Mathematical Analysis from Euler to Riemann. MIT, 1970.)
Ten years later, he published
G. Lejeune Dirichlet. Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält. Abhandlungen der Königlich Preussischen Akademie der Wissenschaften von 1837, 45-81,
where he shows that this behavior is exclusive of conditionally convergent series:
Theorem (Dirichlet). If a series converges absolutely, all its rearrangements converge to the same value.
Proof. Let be the original sequence and a rearrangement. Denote by and their partial sums, respectively. Fix . We have that for any , if is large enough, then for all there is some with . Also, there is a such that for all there is a with , so
Choosing large enough, and using that converges, we can ensure that the two displayed series add up to less than . This gives the result.
In 1853, Riemann proved his rearrangement theorem, although it was not published until 1866, as part of his Habilitationsschrift on representation of functions as trigonometric series, Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe. See here for Riemann’s papers.
Theorem (Riemann). Given any in the extended reals, a conditionally convergent series of reals can be rearranged so that the liminf of the partial sums of the rearranged series is , while the limsup is . In particular, any real can be obtained as the sum of some rearrangement of the original series.
Proof. The idea is very simple, and based on the following observation:
Lemma. If is a sequence of real numbers such that conditionally converges, then, letting and , respectively, denote the subsequence of positive and of non-positive terms, we have that both sequences are infinite, both converge to zero, and for any , we have that .
Proof. The point is that, up to padding the subsequences with zeroes at appropriate places, , so at least one of the two series must diverge, since the original series is not absolutely convergent, by assumption. If one of the series converges, then the other, being a difference of two convergent series, would converge as well. It follows that both series diverge (to ). Since both the and the are subsequences of , which converges to (since converges), then they converge to zero as well. That, in fact, diverges no matter how large is, is immediate.
Using the lemma, we prove the rearrangement theorem in a straightforward fashion. To explain the idea, suppose first that is real. Consider the rearrangement that first adds positive terms until the sum is larger than , then adds non-positive terms until the sum is smaller than , then adds positive terms again, etc, so the partial sums oscillate being larger and smaller than , but each time by smaller amounts, since the and the converge to .
The general case follows the same outline: If is real, define for all . If , define , and if , define . Define the sequence similarly.
We define the rearrangement by stages, and use to denote the partial sums of the rearranged sequence. At the beginning of stage , we have used an initial segment of the subsequence of positive terms, and an initial segment of the subsequence of non-positive terms, and we have defined (where ). At stage , we begin with and . We continue the definition by setting , , etc, until an index is reached such that . Note that exists, since We stop at the least such , and set . Let .
We continue with , , etc, until an index is reached such that . Again, exists because . The least such we call , and this concludes stage .
It is now a routine matter to check that this rearrangement has the desired properties: The sequence of partial sums converges to , because , since . Similarly, the sequence converges to . This shows that the limsup of the partial sums is at least , and the liminf is at most . But the with are increasing, and the with are decreasing, so no larger limit than or smaller than can be achieved.
Corollary. There is an injection of into the set of permutations of the natural numbers.
The idea here is that we can begin with our favorite conditionally convergent series, and assign to the rearrangement with series converging to given by the theorem. Of course there are other methods of establishing the corollary, but I find this argument cute.
Exercise. Modify the construction to show that a conditionally convergent series admits a rearrangement such that any extended real is the limit of a subsequence of the sequence of partial sums.
Prior to Riemann’s theorem, Ohm had obtained interesting related results, that were later extended by Schlömilch, and Pringsheim (see here and here for statements of some of their theorems, and further recent work). They are usually discussed in terms of rearrangements of the alternating harmonic series, although their results are more general. See also these slides by Marion Scheepers for additional work along these lines, and for details see
(In the slides, Scheepers is using the notion of signwise monotonic series. This just means series where is monotonically decreasing.)
Ohm’s theorem appear in
Martin Ohm. De nonnullis seriebus infinitis summandis Commentatio, Berlin, 1839.
It predicts the value of the series obtained if we fix positive integers and , and the terms of the alternating harmonic series are rearranged so that first we add the first positive terms, then the first negative terms, then the next positive terms, then the next negative ones, etc. For instance, if and , the rearrangement is
Theorem (Ohm). The rearrangement of the alternating harmonic series obtained by adding first the first positive terms, then the first negative terms, then the next positive terms, then the next negative terms, etc, converges to .
In particular, for , the limit is and if , the limit is which coincides with the observations we began with.
(See also here.)
[My Master’s student Monica Agana is working on her thesis on the classical theory of rearrangements, covering in detail the results by Ohm, Schlömilch, and Pringsheim.]
A rearrangement is sometimes called simple iff the subsequences of positive terms of the original series and of the rearranged series coincide, and similarly with the subsequence of non-positive terms. Note that the rearrangements in Riemann’s and Ohm’s theorems are simple.
Wacław Sierpiński. Sur une propriété des séries qui ne sont pas absolument convergentes, Bull. Intern. Acad. Sci.: Cracovie A (1911) 149–158,
Sierpiński considers a different kind of rearrangements, where the non-positive terms are all fixed, and only the positive terms are permuted. (The paper is available here.) His result is the following:
- Let be a conditionally convergent series of real numbers that converges to . For any there is a rearrangement of the such that , and the rearrangement leaves fixed all non-positive terms.
- Similarly, for any there is a rearrangement of the such that , and the rearrangement leaves fixed all positive terms.
- Moreover, for any extended real , there is a rearrangement of the such that and, for every , iff .
Before we proceed with the proof, let me mention a related question. Given a conditionally convergent series with sum , it follows from the theorem that if there is a rearrangement fixing non-negative terms and such that , then any numbers in can be so obtained, regardless of whether or not, since a rearrangement of the fixing its non-negative terms is also a rearrangement of the with the same property. Let be the supremum of the numbers obtained via such rearrangements.
Question. Can ? If , is one of these numbers ?
Proof. Note first that it suffices to prove (1), since (2) is an immediate consequence of it: Simply apply (1) to the series , noting that to obtain (2). For (3), we can use (1) or (2) if is real. But it is an easy exercise to show (3) if or .
For completeness, I sketch a proof of (3) when . The case follows the same outline, just as (2) follows from (1). Let and denote, respectively, the subsequences of positive and of non-positive terms of the , so . Split the sequence of into two disjoint subsequences and such that but , and for all .
We describe the rearrangement required in (3) by stages. Our rearrangement fixes all positive terms and only permutes the negative ones. At the beginning of stage , the permuted sequence built so far consists precisely of initial segments of the and of the . In fact, have been used, as have, say, .
Continue by using whenever a non-positive term should be used in the rearrangement, until at least has been used, and a partial sum is obtained that is strictly larger than
(This is possible because while .) Once this happens, use at the first opportunity. This concludes stage .
By induction one easily checks that all partial sums in stage are larger than , and therefore the rearranged series diverges to , as wanted. This completes the proof of (3) for .
It remains to verify (1).
Just as Riemann’s theorem depended on a lemma about the series of positive and non-positive terms of the original sequence, Sierpiński’s result depends on a lemma about the series of positive terms. Unlike Riemann’s theorem, in this case, all the work goes into the proof of the lemma, and the theorem is more properly a corollary of it.
Lemma.Suppose is a sequence of positive numbers such that and Let denote the sequence of partial sums. For any there is a rearrangement with sequence of partial sums such that .
To see that the lemma implies the result, suppose that , and let . As above, let and be, respectively, the subsequences of positive and of non-positive terms of the . Let and denote, respectively, the sequences of partial sums of the and of the . For each , let be the number of positive terms among , and let be the number of non-positive terms, so that , and if denote the partial sums of the , then for all , and .
Apply the lemma to the to obtain a rearrangement with partial sums such that . Consider the rearrangement of the that fixes the and permutes the as indicated. If are the partial sums of the , note that, by design, , so that , and , as wanted.
Note that the requirement that in the statement of the lemma, and therefore that in (1), cannot be improved in general to cover a larger range of possible values, since the may be decreasing, in which case we have that for all .
All that remains is to prove the lemma.
Proof. As in Riemann’s theorem, we proceed by stages. Starting with , at stage we examine to decide the value of . As in that theorem, we want to arrange that the values increase if smaller than , and decrease if larger, so that in the limit we obtain the desired value. Actually, we will need to modify slightly this strategy. To motivate the proof, consider first the case where the are monotonic, that is, In this case, the strategy works: At stage , we consider two cases, according to whether or :
- If , then let be such that , and set . This is possible, since .
Note that, for as long as , we stay in this case. Thus, if from some point on we are always in case 1, then for large enough we have
This is a contradiction, and indicates that repeated applications of this case always terminate, and lead to case 2. In particular, case 2 is considered infinitely often. Moreover, if is least such that , then .
- If , then in particular , so at least one of the , , has not been considered yet as value of a Pick the least such index , and define .
Since this case applies infinitely often, the value of the least index such that is not one of the increases unboundedly. By design, no index is used more than once, and therefore this algorithm results in a rearrangement of the . If we ever find an index such that (again) , note that .
Note that our assumption that the are decreasing ensures that, in case 2, (with notation as above) , since . Moreover, it cannot be that we have equality from some point on, since the converge to . This means that the values decrease.
Since , it follows that, if case 1 is also applied infinitely often, then . Therefore, to conclude, it suffices to argue that we cannot stay in case 2 forever. To see this, argue by contradiction, and assume that from on, we always stay in case 2. Let be the number of indices such that is not one of the , . For any , since , then , with equality if and only if is not one of the , . Since , necessarily at least one of the , , is for some If is the least such index , then , because is one of the with , in fact . Since we cannot have an infinite decreasing sequence of positive integers, this means that (we have a contradiction and) eventually we should be in case 1 again. This completes the proof in the case the are decreasing.
Let’s now consider the general case. If we try to implement the argument we just described, we see that if we are ever in case 1, , the sequence of increases until we find an index with . Also, once we enter case 2, we cannot stay there indefinitely, as the number of terms that are not some , , decreases. The problem is that the sequence of is not necessarily decreasing. In fact, if we are at a stage where we define for some , and it happens that , then . This means that, although we have ensured that , and that there is a subsequence of the converging to , it may well be the case that .
Sierpiński deals with this situation in a clever fashion. He ensures that if we are in case 2, so that for some , then also for some , which means that was defined according to the prescription in case 1. He uses this to guarantee that cannot stray too far from . In detail, Sierpiński proceeds as follows, defining now three cases. Assume we are at the beginning of stage :
- If , as before let be some where the index has not yet been chosen, and , and
- If , now we consider two possibilities, according to whether or not the index was picked previously: If it was not, then we let . Note that if this is the case, and that, since , then necessarily some with must be for some . This means that at some later stage (at most by stage ), we are no longer to be in this case.
- If , and the index was chosen previously, then we let be , where is the first index less than not chosen yet.
As before, cases 1 and 3 happen infinitely often, so we have indeed defined a rearrangement. Moreover, if we are in case 3, then for some (this is why we included case 2). The point is that if stage is by case 2, then , and if it is by case 3, then for some . This means that the only way we can have for is if stage was by case 1, which means that .
Fix . Since cases 1 and 3 happen infinitely often, if we choose large enough, then we can ensure that all indices mentioned below are so large that , and if for some , then where . We want to ensure that . This proves that .
If stage is by case 1, and stage was largest where , then . Since the sequence is increasing for , we have , as wanted.
Suppose then that stage is by case 3, and that stage was largest where we were in case 1. We then have that . Also,
but for any such , we have that stage was either by case 2, and therefore , or else it was by case 3, and therefore for some , but necessarily , so
and stage was by case 3 and stage was by case 3
This means that . But also
This completes the proof.
This concludes the proof of Sierpiński’s theorem.
Further work on rearrangements that fix pointwise prescribed subsets of has been pursued by Wilczyński and others, partly in the context of descriptive set theory. See for instance
Rafał Filipów, and Piotr Szuca. Rearrangement of conditionally convergent series on a small set. J. Math. Anal. Appl., 362 (1), (2010), 64–71. MR2557668 (2010i:40001).
Parts of this post have been used in this MO question.