SUMMABILITY OF ALTERATIONS OF CONVERGENT SERIES

The effect of splitting, rearrangement, and grouping series alterations on the sun=nability of a convergent series by l-l and cs-cs matrix methods is studied. Conditions are determined that guarantee the existence of alterations that are transformed into divergent series and into series wth preassigned sus. KEV WORDS AND PHRASES. l-1 mehod, cA-cA mthod, rearrangement, spting. 1980 MATHEMATICS SUBJECT CLASSIFICATION CODES. P 40C05, Secondary-40A25.

T.A. KEAGY apq converges for each q and pq p,q-i ) and Lorentz [2 have characterizedmethods A by the

Knopp property
SUpq.p[apq < (R).Each of the above methods is limit-preserving (qXq .p(AX)p)whenever a 1 for each q.P Pq (l.1) The main purpose of this paper is to determine what effect alterations of a convergent series may have on the summability of the series under methods of the above type.In 2, we describe each of the alteration types and investigate properties of the alterations and their respective matrix representations.In 3, alterations of convergent series that map to divergent series by cs-cs or transformations are determined.Finally, 4 is concerned with alterations hat are mapped by d-c orransforations to series that sum to a pre- assigned value o.

ALTERATIONS OF SERIES.
By a grouping alteration of a series .ixiwe will mean a series iYi determined by an increasing sequence of positive integers {k(i)}i-i where the first term of the altered series is the sum of the first k(1) terms of the original series, and for i > 1 k(i) Yi LJfk(i-1)+ixj" Grouping alterations may b.e written in matrix form as transformations with all entries 0 or 1 that satisfy all requirements for limit-preserving cs-cs andmethods.
Rearrangements are perhaps the most familiar type of series alteration.
They may be represented in matrix form as transformations with all entries 0 or 1 which have exactly one nonzero entry in each row and each column.Such maps are easily seen to be limlt-preservlngtransformations, but they need not nec- essarily be cs-cs.The answer to the question of precisely which rearrangements preserve the limlt of all convergent series dates at least as far back as 1946 when Levi [3 first established that a rearrangement p(1), p(2), p(3), of the positive integers will always yield a convergent rearranged series ap(k) la i whenever a i is convergent if and only if there exists an integer N such that for each m the set of integers p(1), ... p(m) can be represented as the union of N or fewer blocks of consecutive integers.Subsequent proofs of the same result by Guh here since he utilized (i.i) and (1.2) in his proof, a technique we will use in the proof of our Theorem 2 later.
Splittings of series were introduced by P. Wuyts [6].If .iai is a series, then for each i we write a i a(i,1) + + a(i,ki).The resulting series a(l,l) + + a(l,kl) + a(2,1) + + a(2,k 2) is a splitting of the original series.In one way a splitting may be thought of as being the opposite of a grouping alteration since a grouping alteration produces a new series with sequence of partial sums a subsequence of the original series sequence of partial sums and a split series produces a supersequence of the original sequence of partial sums.The following theorem provides even more insight into the connection between these wo types of alterations.
THEOREM i.Let xi and Yi be wo series.There exist a splitting ai PROOF.We first determine a splitting ai of xi by letting a I YI' a2n x and i, n a2n-l a2n+l Yn+l a2n for n 2 3 Note that a2n_l + a2n Xn for each n.We now determine a grouping alteration libi of .iaiby letting b I a I and bn+I a2n + a2n+l.Than b I a I Yl and bn+I (Xn a2n-l) + (Yn+l a2n) Yn+l for n > i, hence the proof is complete.
Representations of spllttings in matrix format presents some difficulty if the series contains zero terms.If the series does not contain any zero terms, (2.1) may be expanded as where al(b(l,l) for each i.The splitting may now be represented in matrix form as a series to series transformation where each row has exactly one nonzero entry.The first k I entries of the first column will be b(l,l), b(l,k I), and the k I + i through the k I + k 2 entries of the second column will be b( The remaining columns are formed similarly.SUPn(lb(n,l) + ..-+ lb(n,kn )l) < oo.
Furthermore, any splitting matrix that is cs-cs oris also limit-preserving.
This yields a result reminiscent of, yet distinctly different from, the following theorem due to .Wuyts [6].
THEOREM.(Wuyts) The split series (2.1) of a convergent series is itself convergent if and only if limnmaxksk la(n,l) + + a(n,k) O.In [7] we provided an affirmative answer to the following question proposed by J.A. Fridy [8]: is a null sequence necessarily in if there exists a sum- preserving 4-4 matrix that maps all rearrangements of x into 4? A similar ques- tlon is as follows: if x e cs (4) and A is a limit-preservlng cs-cs (4-4) matrix, does there exist an alteration of x that A fails to map into cs (4)?
Grouping alterations are always limlt-preserving cs-cs and 4-4 transfor- mations, therefore it is easy to see that if A is a limit-preserving cs-cs (4-4) matrix and x cs (4), then A will map every grouping alteration of x into cs Rearrangements are also limlt-preserving 4-4 transformations, therefore it follows that A wll map every rearrangement of x into whenever A is 4-4.
Rearrangements are not necessarily cs-cs as noted above.The following theorem resolves the question in-case x cs and A is cs-cs.
THEOREM 3. Let A be a limit-preserving cs-cs matrix and x cs such that x .There exists a rearrangement y of x such that Ay cs.PROOF.By (1.2) it is clear that each row of A is of bounded variation and hence is convergent to some L. Suppose row p of A converges to L 0. We now construct a rearrangement y of x such that qapqyq fails to converge.Suppose Let Yi zi for m < i < k and continue this building process to obtain a re- arrangement y of x such that for each N there exist n,m > N such that [q=napqyq[ > i. Suppose now that each row of A is null.Let {y(l,J)}-i be a rearrangement of x such that qy(l,q) 2 and n sup n[.l y(l'q) < M 3(suPilXi[ + i).
This selection process may be continued for y.Since SUPn[nq=lyql < M, it follows that supn,mlq=nyq] <_ 2M.Therefore by the pattern established by (3.2)   and (3.5), qapqyq will converge for each p.It also follows from (3.1) and (3.4) that the selection process for y may be accomplished so that IV p(2n) (Ay) r p(2n+l) (Ay) > l "p=l p "p=l p for n 0, I, 2, 3, Hence A fails to map y to cs, and the proof is complete.
Splittings need not be cs-cs or 4-maps.The following theorem leads to an answer to the question as to whether a limit-preserving cs-cs (4-4) matrix nec- essarily maps some splitting of every series x in cs (4) into a series not in cs (z).
THEOREM 4. Let A,be.a.matrix with an infinite number of nonzero columns, each one of which is in cs.If x is a sequence, then there exists a splitting y of x such that Ay is not null.
COROLLARY 5.If A is a limit-preserving cs-cs (E-E) matrix and x cs (E), then there exists a splitting y of x such that Ay fails to be in cs ().
Rearrangement is the only series alteration method of our three types that can produce a new series with sum different from that of the original series.A more interesting question is the following: if A is a limit-preserving cs-cs matrix and o is a preassigned value, under what ,conditions can a convergent series be altered so that A maps the altered series to one that sums to o?
The answer for grouping alterations is easy since groupings are limit- preserving cs-cs and E-maps: the alteration exists only when is the sum of the original series.The same answer applies when A is E-E, x , and the alteration is rearrangement.When A is cs-cs, x cs but x 4 , and the altera- tion is a rearrangement, the desired alteration will always exist.
The answer for splitings is more complex and in fact depends on the partic- ular matrix A in question.If the limit-preserving cs-cs (E-A) matrix is equivalent to convergence it is clear that A can map a splitting of the original series to a series with sum s only if u is the sum of the original series.THEOREM 6.Let A be a matrix with null rows and an fncreaslng sequence of columns {q (i) }iffil such that i for i i, 2, 3, and n m{p Y.p_-(a p ) I} 0 p,q(21) ,q(21-1) n If x is any sequence wlth {If=ixi} i bounded and o is any number, then there exists a splitting y of x such that Ay cs with sum .

THEOREM 2 .
The splitting matrix B is cs-cs if and only if SUPn,kk Ib(n, I) + + b(n,k) < =, n and B isif and only if

n 3 .
MAPPINGS TO DIVERGENT SERIES.
the first n terms of y have been determined.Let K i + ILI(I + maxilxil) and m > n + i such that m.Choose -Y xjj where j min{q: x is not one of -xY' Yi_l }_ for q n < i < m.Rearrange the terms of x not included in YI' "''' Ym-m such that ,lY+/-z+/-l > 2/IL I. Then k epq l.mz 2) > p(1) such that if t > p(2), then )I < 2-2/(Io-I + ).