JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 225, 532᎐541Ž. 1998 ARTICLE NO. AY986046 Formal Summation of Divergent Series Robert J. MacG. Dawson* Department of Mathematics and Computing Science, St. Mary's Uni¨ersity, Halifax, No¨a Scotia, Canada B3H 3C3 View metadata, citation and similar papersSubmitted at core.ac.uk by H. M. Sri¨asta¨a brought to you by CORE Received August 18, 1997 provided by Elsevier - Publisher Connector The idea of telescoping a series is widely known, but is not widely trusted. It is often treated as a formalism with no meaning, unless convergence is already established. It is shown here that even for divergent series, the results of telescop- ing are self-consistent, and consistent with other well-behaved summation opera- tions. Moreover, the summation operations obtained by telescoping are the strongest possible operations with these properties. Some Tauberian theorems are exhibited for telescoping. ᮊ 1998 Academic Press 0. INTRODUCTION The technique of ``telescoping'' series is well known, and where applica- ble is very fast and intuitive. A curious and well-known feature of this technique is that it provides a ``sum'' for many oscillatory and divergent series as well as for many convergent series. Hence, students are often cautioned not to use it without first verifying the convergence of the series analytically. This is to avoid such ``results'' as Ž.1 q 2 q 4 q 8 q иии y 21Ž.q 2 q 4 q иии s Ž.1 q 0 q 0 q 0 q иии « Ž.1 q 2 q 4 q 8 q иии sy1. Ž 1. This summation, as Bellwx 2 observes, may also be obtained from ``the 1 formal binomial theorem applied toŽ. 1 y 2y .'' He then goes on to describe it as ``a meaningless result that did not astonish Euler,'' obtained ``without sufficient attention to convergence and mathematical existence.'' * Supported by NSERC Grant OGP0046671. 532 0022-247Xr98 $25.00 Copyright ᮊ 1998 by Academic Press All rights of reproduction in any form reserved. FORMAL SUMMATION 533 In defense of Euler, it should be noted that, although rather alien to real analysis, this equation holds in the field of dyadic numbers, and in the twos-complement arithmetic used by many computerswx 1 . The summation method of Tommwx 8 also sums every power series Ý r i to its formal sum 1r1 y r. It is the purpose of this article to show that telescoping can be put on a rigorous foundation, and its consistency with other methods of summation guaranteed. 1. DEFINITIONS Let R be a ring with identity 1. A series over R is a vector with elements in R indexed by the natural numbers; it is distinguished from a sequence by the choice of product. Products of sequences are generally defined termwise, reflecting their comparatively weak natural order struc- ture; in contrast, the vector space of series is usually made into an R-algebra using the Cauchy product. This algebra has multiplicative iden- tity 1 s Ž.1 q 0 q 0 q иии . The Cauchy product preserves the right-shift operation ␴ , in the sense that ␴A) B s A)␴ B s ␴ Ž.A) B . The right-shift operation is internally represented by the elementŽ. 0 q 1 q 0 q 0 q иии s ␴ 1 of the algebra of series over R; henceforth, this series is also represented as ␴ . The algebra of series is, of course, isomorphic to the algebra R@␴ # of formal power series over R. We shall sometimes write series as power series; thus the same series may be written asŽ. 1 y 1 q 1 y 1 q иии ,or 23 1 y ␴ q ␴ y ␴ q иии . The subalgebra generated by ␴ is Rwx␴ ,the algebra of polynomials. An element of Rwx␴ may be written omitting terminal 0s; so instead ofŽ. 1 y 2 q 0 q 0 q иии we write 1 y 2␴ . Following axioms A, B, and C of Hardywx 6, p. 6 , we define a summation to be a linear function ᑭ: A ª R, where A is a vector subspace of R@␴ # and is closed under ␴ , ᑭŽ.1 s 1, and ᑭ Ž␴ X .s ᑭ ŽX .. It is not hard to show that any summation takes a finitely supported series to the sum of its nonzero terms. If ᑭ, ᑭЈ are summations, domŽ.ᑭ : dom ŽᑭЈ ., and ᑭЈ NsdomŽᑭ . ᑭ, then we call ᑭЈ an extension of ᑭ and we write ᑭ : ᑭЈ.If ᑭ12and ᑭ agree on the intersection of their domains, then we call them compatible. It is easily shown that: PROPOSITION 1.1. Two summations are compatible if and only if they ha¨e a common extension. For instance, it is known that Euler's transformation methodw 5, Section 63; 6, Chap. VIIIxw and the CesaroÁÈ᎐Holder method 5, Section 60; 6, Chaps. V and XIx are compatible, but each sums some series that the other does not. From the preceding proposition it follows that they have a common proper extension. 534 ROBERT J. MACG. DAWSON If a summation is compatible with every summationŽ resp., extension of ᑭ.Ž.we call it canonical resp., ᑭ-canonical . For any ᑭ,theᑭ-canonical summations are filtered by the extension relation. Thus, there is a unique maximal extension that extends every canonical extension of ᑭ. This will be called the closure of ᑭ. A series is in the domain of this closure if and only if there is a unique value to which a summation compatible with ᑭ can sum it. Naturally, a summation which is its own closure is called closed. Familiar examples of summations include the finite summation ᑛ, whose domain is the algebra Rwx␴ of series with finite support; the standard summation ᑭ a on the algebra of absolutely convergent series; and the standard summation ᑭ c on the vector space of convergent series. By definitionwx 6, p. 10 a regular method of summation is an extension of ᑭ ca. Note that both ᑭ and ᑭ cmay be factored into the form ⌳⌺ where ⌺ is the linear operator on R@␴ # that takes a series to its sequence of partial sums, and ⌳ takes such a sequence to its limit. ⌺ is represented 1 internally within R@␴ # by the series ⌺ s Ž.Ž.1 q 1 q 1 q иии s 1 y ␴ y . The standard summation of convergent series over the field ZÃ p of p-adic numbers is also a summation of this type. Many other examples of summations, which can sum some divergent series, are given in Hardywx 6 . Some are based on generating functions, such as the methods of Abel and Lindelof.È Others, such as the Y method or the CesaroÁÈ᎐Holder method, use appropriate linear operators to im- prove summability, applied either to the series itself or to the sequence of partial sums. One important class of summations is the Nùrlund means. Let P be a series with terms piP. For any sequence S,let N Ž.S be the sequence whose terms are Ž.Ž.P )S iir⌺ P ;if⌳ N P⌺ Ž.A exists and equals a, we define ᑭ PPŽ.A s a. A Nùrlund mean ᑭ is regular iff pirŽ⌺ ŽP ..iª 06, Žw Section 4.1x.Ž . Any two regular Nùrlund means are compatible; so by Proposition 1.1, there is a universal Nùrlund summation ᑭ N that extends all regular Nùrlund means.. If P is ᑭ-summable and does not sum to 0, we may define TP Ž.A to be the series whose terms are Ž.Ž.P ) A rᑭ P .If ᑭ is a limit of partial sums, this is closely related to NP . PROPOSITION 1.2. Suppose that P g domŽ.ᑭ where ᑭ s ⌳⌺, and either ⌳ preser¨es products or P is finitely supported; then ⌳⌺ TPPŽ.A s ⌳ N ⌺ Ž.A Proof. By definition n apjk Ž.⌺ TP Ž.A n s T ÝÝᑭ P is0 jqksi Ž. 1 j s apjk ᑭ P ÝÝ Ž.iqjsnks0 FORMAL SUMMATION 535 1 nnyk s pakj ᑭ P ÝÝ Ž.ks1 js0 ⌺Ž.P n s Ž.NP ⌺Ž.A n .2Ž. ᑭŽ.P N If P is finitely supported, ⌺Ž.P n s ᑭ Ž.P for large enough n.If⌳ preserves products, then ⌺Ž.P ⌳⌺ TPPŽ.A s ⌳ N ⌺Ž.A T ž/ᑭŽ.P N ⌺Ž.P s ⌳⌳Ž.NP ⌺Ž.A ž/ᑭŽ.P N s ⌳ NP ⌺Ž.A .3Ž. In the case where P is finitely supported, this is the familiar process of telescoping a series, which we consider in the next section. If P is not finitely supported, but ⌳ preserves products, TP yields a generalization of telescoping which will be considered in a sequel to this article. 2. THE TELESCOPIC EXTENSION A summation is not required to preserve products; for instance, ᑭ c sums S s Ž.1 y 1r '''2 q 1r 3 y 1r 4 q иии but not S)S. However, even if a summation does not preserve products, linearity and shift-inde- pendence imply a weak multiplicative property. PROPOSITION 2.1. For any summation ᑭ o¨er R, any element X of domŽ.ᑭ , and any element F of Rwx␴ , F) X g domŽ.ᑭ , and ᑭ ŽF) X .s ᑭŽ.F ᑭ ŽX .. Multiplying a series by an element of Rwx␴ is the same thing as telescoping it: shifting the series to the right one or more times, multiply- ing the various shifted series by scalars, and adding. This is often done to obtain a series whose sum is known, from which a sum for the original series may be inferred; the next definition makes this rigorous.