Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at

A Note on λ-Permutations

Author(s): Daniel J. Velleman

Source: The American Mathematical Monthly, Vol. 113, No. 2 (Feb., 2006), pp. 173-178

Published by: Mathematical Association of America

Stable URL:

http://www.jstor.org/stable/27641871

Accessed: 25/01/2010 13:59

Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at

http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless

you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you

may use content in the JSTOR archive only for your personal, non-commercial use.

Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at

http://www.jstor.org/action/showPublisher?publisherCode=maa.

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed

page of such transmission.

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of

content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms

of scholarship. For more information about JSTOR, please contact support@jstor.org.

gis) = cos s ? cosias) cosibs),

then gw(0) = g"iO) = g'(0) = giO) = 0, while g""i<d) = -4a2b2.

Taking a ? 1/2 and b ? V3/2, it follows that there exists S > 0 such that 0 = g(0) > g(2?) when 0 < 0 < 8. This gives cos0 = cos(2?)/cos(9 < _cosiv^?), whence _{0 > \/3#} as claimed.

Incidentally, we remark that simple modifications of these arguments (including the use of the hyperbolic cosine rather than the cosine in the calculus lemma) yield a proof

that the hyperbolic plane is not (locally) flat.

Department of Mathematics University of Florida Gainesville, FL 32611-8105

robinson @ math. ufl. edu

A Note on ?-Permutations

Daniel J. Velleman

In their paper [3], Steven Krantz and Jeffery McNeal study what they call X-permut ations. They define a permutation a of the positive integers to be a ?-permutation if it has the following two properties:

1. For every convergent series Y1T=\ ak>trie series Y1T=\ ae(k) also converges. 2. There _{is at least one divergent series Y1T=\ a^ suc^ ^at 1lT=\ a^(?) converges.}

We call a permutation a with property 1 a convergence-preserving permutation. Notice that property 2 is equivalent to the statement that a-1 is not convergence-preserving. Thus, a permutation a is a ?-permutation if and only if it is convergence-preserving, but its inverse is not.

We _{begin with} a characterization _{of convergence-preserving permutations.} The characterization _{we give can be found in [5], where it is derived by combining results} of R. P. Agnew [1] and P. Vermes [7], but our derivation is more direct. A similar characterization _{can also be found in [4], and related results can be found in [2] and}

[6].

We will need the following notation: if c and d are positive integers and c < d, then

[c, d]z = {x e Z+ : c < x < d}.

Suppose now that a is a permutation of the positive integers. For any positive integer n we can write _{{<r(l), or (2), ..., a in)} as a union of blocks} of consecutive _integers. More _{precisely, we have}

{ail), a (2), ..., a in)} = [cnx, d?]z U [cn2, dn2}z U U _{[cnbn, dnbJz,} (1)

where c" and d" (1 < / < bn) are positive integers, c" < ?/f, and c"+1 >

please by choosing n sufficiently large. Let

Mn = _dnbn + 1 = max{or(l), or(2), ..., a(n)} + 1.

We _{say that bn, the number of blocks in the union in (1), is the nth block number for a} and that {bn}%L{ is the block number sequence for a. Finally, we call a sum-preserving

if for every convergent series Y1T=\ a^ tne sei"ies YlkLi a?(k) a^so converges, and the two series have the same sum. With _{this terminology in place, we can now state our}

characterization _{of convergence-preserving permutations.}

Theorem _{1. The following} statements _{concerning a permutation a of the positive} in

tegers are equivalent:

(a) a is convergence-preserving.

(b) The block number sequence for a is bounded.

(c) a is sum-preserving.

Proof. Clearly (c) =>> (a); we prove that (a) =^ (b) and (b) => (c).

(a) => (b): Suppose that the block number sequence for a is unbounded. We will find a convergent series Y1T=\ ak sucn tnat YlkL\ a?(k) diverges.

Choose _{rt\ such that c\x} ? 1. Now _{define ak for k satisfying} 1 < k < _Mni as fol lows:

1 if k ? d"1 for some / with 1 < / < _bn{,

ak ? _{\?\ if k = d"1 + 1 for some / with 1 < / < bn],} 0 otherwise.

Notice that

Y^acr(k)

=

bnx

> 1,

YLUk

= 0'

k=\ k=\

while _Yll=\ ak is either 0 or 1 when 1 < n < _Mn{.

Proceeding recursively, we will now choose for each j > 1 a positive integer n? such that Mn. > Mn._x, and we will also define ak for k satisfying Mn._x < k < Mn.. At each step, we will make our choices so as to ensure that

Mn:

Y^ak=0

Suppose that j > 1, that a positive integer n}_\ has been chosen, and that ak has been chosen for k satisfying 1 < k < Mn._x in such a way that

Mn._x

^2 ak

= 0.

?:=l

ak =

1/j ifk = d"J for some / with 1 < / < bn., ?

1/j if k = d\J + 1 for some / with 1 < i < bn., 0 otherwise

Then

Mflj j

bn.

^ao{k)

= ? > _;, ^fljk

= _0,

?:= 1 7 ?=1

and _{XX=i ?it is either 0 or 1/j when Mn. _1 < n < Mn..}

This _completes the definition of the series

E?lia?- Clearly J2T=\ ak ? 0, but ES=i a?(k) diverges.

(b) => (c): Suppose that the block number _sequence for a is bounded and that

Y^Li ctk ? L. For each positive integer n letS(n) = E*=i ?jfe.Thenlim?^00 Sin) ? L. For any n sufficiently large that c\ ? 1, we have by equation (1)

dn u dn u

?aff(i)

=

?flJt

+

? ?flt

=

5K)+?(swr)

-

s(c"

-

!))

(2)

it=l fc=l /=2 yt^J2 /=2

Now d\ -? oo as ^ -> oo, so all of the values of S in the last sum in equation (2) approach L. Since bn is bounded, it follows that

Eaa(k)

= _lim

y^^a(^ = L,

n->oo J

as required.

Theorem 1 resolves two issues from _{[3]. The first is that on page 33 Krantz} and McNeal _suggest that it would be of interest _{to investigate the sum-preserving}

?-permutations as a subset of all ?-permutations. However, according to Theorem 1, this subset is in fact the entire set of ?-permutations: all ?-permutations are sum preserving. Second, it is evident from equation (2) that if a is convergence-preserving,

then there is a positive integer B such that, when the terms of a series are permuted by a, every partial sum of the permuted series can be written as a linear combination with coefficients ?1 _{of at most B partial sums of the original series. In other words,}

letting Sin) denote the nth partial sum of the original series, for each n there is a positive integer j satisfying j < B, a sequence of positive integers m\, m2, ..., m;-, and a sequence of coefficients _{c\, c2, ..., cj; in {? 1, 1} such that}

n j

^2a?(k)

=

^CiSimi).

We can take B to be twice the bound on the block number sequence for a. This shows that the example in the second paragraph of section 5 of [3] is incorrect.

In section _{3 of [3], Krantz and McNeal show that there are uncountably many}

?-permutations, but they do not compute the exact cardinality of the set of ? permutations. In fact, this is not hard to do. As in [3], we let J\f be the set of all ?-permutations, and ? the set of all permutations a such that both a and a~x are

of the positive integers that swaps 2m ? 1 and 2m if m is in X and leaves 2m ? 1 and 2m fixed otherwise. _{It is easy to see that for every n the nth block number for rx is} either 1 or 2, and therefore xx is convergence-preserving. It is also clear that xx is its own inverse, so xx belongs to ?. Furthermore, the permutations rx for different X are different, so this is a family of 2K? distinct elements of ?. Thus, card(?) > 2^?. But

the set of all permutations of the positive integers has cardinality 2K?, so it follows that card(O) < 2^? as well, and therefore card(O) = 2*?.

Now _{let a be any ?-permutation. It is easy to see that the composition} of two

convergence-preserving permutations is convergence-preserving, so xx o a is conver

gence-preserving for every set X of positive integers. If (xx o a)"1 = o~~l o xx were

convergence-preserving, then a~x o rx o xx = a~l would be as well, contradicting the

fact that a is a ?-permutation. Therefore rx o a is a member of ?? for every X. Clearly these are all distinct, so they are 2^? distinct elements of ??. As before, it follows that card(A0 = 2K?. Thus, we have established the following theorem:

Theorem _{2. card(AT) = card(O) = 2^.}

Finally, we give examples of two infinite series whose behavior with respect to ?-permutations is interesting and state an open problem motivated by these examples. The _{terms in both of these series are the numbers ?1, ?1/2, ?1/3,...,} but the

terms are arranged in different orders in the two examples. In both series, the pos itive terms appear in the order 1, 1/2, 1/3,... and the negative terms in the order ?

1, ?1/2, ?1/3,...; the only difference between the series is in how the positive and negative terms are interleaved.

For the first example, we alternate blocks of positive terms with individual negative terms, and we make sure that each block of positive terms adds up to at least 2. Thus, the series begins like this:

?

111

11

11

fi

234

56

33

2

The sequence of partial sums for this series goes up by at least two for each block of positive terms and down by at most one for each negative term. It is clear, therefore,

that the partial sums approach infinity, and the series diverges. Of course, the terms can be rearranged to make the series converge; we might say that it is a "conditionally

divergent series."

Now, consider the result of permuting the terms of this series by some ?-permutation a. Let B be a bound on the block number _sequence for a. It is not hard to see that

the sum of a consecutive _sequence of terms in this series cannot be smaller than ? 1.

Therefore, as in equation (2), we obtain for sufficiently large n

d'{ bn df d? d?

^aa{k)

=

^ak

+

Y^Ylak

-

Ylak

~

(bn

~

_{^ - XIa*}

~~

(B

-

!)

k=c^ k=\ k=l

Because _{dnx ?> oo as n -> oo, the partial sums of the permuted series also approach}

infinity. This shows that, although the original series can be made to converge by per muting the terms, this cannot be done by using a ?-permutation.

where _{t\ > s\ + 1, the next block is positive terms adding up to s2, where s2} > t\ + 1,

and so on. Thus, the series starts like this:

?

11111

11

> ak = 1 - 1-1-1-1-1-.

f?

2

3

423

33

5

The partial sums of this series first rise to s\ > 1, then fall to s\ ? t\ < ?1, then rise to Si ? t\ + s2 > si + 1 >2, then fall to s\ - t\ + s2 ? t2 < ?2, and so on. Clearly this

is another conditionally divergent series.

Now _{suppose that L is any real number. We will find a ?-permutation} a such that, when _{the terms of the series are permuted according to a, the sum of the resulting}

series is L. We begin by finding a number m \ such that ami > 0 and

m\

L <

y^aic

< L + 1.

Such an m\ must exist because of the way the partial sums swing back and forth between _{larger and larger magnitude positive and negative numbers and the fact that} all terms in the series have magnitude at most _{1. The first m\ terms of the permuted} series are the same as those of the original series. In other words, we let o{k) ? k when k <m\. Suppose that am[ is part of the jth positive block, whose sum is Sj. For

term number mi + 1 of the permuted series, we use the first unused negative term of the original series; this is the first term of the 7th negative block, whose sum is ?tj. We then continue _{choosing terms of the permuted series according to the following} rule: if the sum of the terms chosen so far is less than or equal to L, then we choose the first unused positive term to be the next term of the permuted series, and if it is greater

than L, then we choose the first unused negative term. Notice that this is exactly the same procedure that is usually used to prove that a conditionally convergent series can be rearranged to have any desired sum. As in that proof, it is clear that the resulting permuted series has sum L and that

n

L - 1 <

^aa(k) < L + 1

k=\

if n > mx. What _{remains to be shown is that the permutation o generated in this way}

is a ?-permutation.

Notice _{that in the permuted series the first term of the j th negative block is used at}

step m\ + 1, which occurs before the step at which the first term of the (7 + l)th pos itive block is used. Also, the first m\ terms of the permuted series include all terms of the original series before the 7 th positive block, as well as some terms of this positive block.

We _{claim now that in the permuted series the first term of the (7 + l)th positive} block is used before _{the first term of the (7 + l)th negative block. To see why, suppose} not. Then the first term of the (7 + l)th negative block is used first, say as term number mr + 1 of the permuted series. This means _{that for k satisfying mx < k < m' term}

be greater than L. Consequently, we have

m m\

L < ^aa[k)

<

^2aa(k) + s, ~

tj < L + 1 - 1 = L,

k=\ k=\

which is a contradiction.

Thus, the first term of the (j + l)th positive block is used at some step m2 + 1 that comes before the step at which the first term of the (j + l)th negative block is used. Moreover, the first m2 terms of the permuted series include all terms of the original

series before _{the y th negative block, as well as some terms of this negative block.} Similar reasoning can be used to show that the first term of the (j + l)th negative block is used before the first term of the (j + 2)th positive block. Indeed, by induction we can demonstrate _{that the first term of every block is used before the first term of}

the next block. Since both the positive terms and the negative terms are used in order, it is not hard to see that the block number sequence for the permutation a is bounded by 2. Thus, by Theorem 1, a is a ?-permutation.

These _{examples suggest the following question, to which we do not know the an} swer. Suppose that Y1T=\ ak *s a conditionally divergent series. We know that this series can be rearranged to converge to any real number. But what if we restrict our attention to ?-permutations? Let

S = \ L e R _{: for some ?-permutation a,}

V] ao{k) = L \ .

[

k=\

J

We have seen an example in which S ? 0, and another in which S = R. Can S ever be anything in between?

REFERENCES

1. R. P. Agnew, On rearrangements of series, Proc. Amer. Math. Soc. 6 (1955) 563-564. 2. _{U. Elias, Rearrangement of a conditionally convergent} series, this Monthly 110 (2003) 57. 3. S. G. Krantz and J. D. McNeal, Creating more convergent series, this MONTHLY 111 (2004) 32-38. 4. R A. B. Pleasants, Rearrangements that preserve convergence, J. London Math. Soc. 15 (1977) 134-142.

5. P. Schaefer, Sum-preserving rearrangements of infinite series, this Monthly 88 (1981) 33-40.

6. J. Stef?nsson, Forward shifts and backward _{shifts in a rearrangement of a conditionally convergent} series, this Monthly _{111 (2004) 913-914.}

7. P. Vermes, Series-to-series transformations _{and analytic continuation by matrix methods,} Amer. J. Math. 71(1949)541-562.

Department of Mathematics and Computer Science Amherst _College