Vol. 10 No 7 - Pi Mu Epsilon

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550 PI MU EPSILON JOURNAL By proceeding in the manner described in the preceding paragraph, the follo\ving rearrangement of the given series is obtained: 1 1 1 l -1 (l '0)-2 ( 1. 0)-3 (l' 0)-4 ( 1 '0)- 1 ( -1.-1) -2 (-I' -1)- 1(0, 1) - _!_( l , 0) - _!_( 1, 0) - _!_( 1 , 0) - _!_( 1, 0) - _!_( 1 , 0) - _I (1 , 0) - - 1 ( 1, 0) 5 6 7 8 9 10 11 l 1 l l 1 1 -3( -1, -1)- 4( -1, -1)- 2(0, 1) -12(1,0) -13(1,0) -14(1,0) 1 I l I l -15(1,0)-16(1,0) -17(1,0) -18(1,0) -19(1,0)- ···, which appears to be converging to S = (-1, 1). Levy then uses a complicated geometric approach to show that the rearranged series actually converges to S = (-1, 1). In the last paragraph of his paper, Levy claims that his method may be generalized to consider rearrangements of infinite series of vectors in n dimensional space [6, p. 511]. His main idea is to decompose the series under consideration into n + 1 partial series. This corresponds to the idea of decomposing a series of complex numbers, or equivalently a series in R 2 , into three partial series. Levy does not, however, elaborate on exactly how this procedure would work. Levy's article is very concise. It offers neither proofs nor examples, and is somewhat poorly written. Knopp [5, p. 398] and Kadets [4, p. 1] state that Levy proved the result on the description of the set of all rearrangements of a series of complex numbers but they make no mention of the fact that he at least conjectured the result for n-dimensional space. Rosenthal claims [9, p. 342] that it was Levy who first proved the result for n-dimensional space. The fact is that Levy did not prove the result, but he was at least aware of it. III. Steinitz' Results. Steinitz begins the introduction to his paper on conditionally convergent series in convex systems by briefly tracing the history of the study of the rearrangement of series of real numbers. He introduces the term "summation range" of a series. RANGEMENT OF SERIES, SMITH 551 finition 1. Let X be a finite dimensional vector space. The summation 00 of the series L ~ where ~ E X, is the set of all vectors x E X such that k=l L "-..(k> converges to x where L "-n(k> is a rearrangement of L xk. t:l k=l k=l Steinitz then restates Riemann's result as follows: The summation range of conditionally convergent series of real numbers is the set of all real numbers [10, p. 128]. Next he considers the summation range of a complex series ~nd gJ\'es two brief examples in the complex plane [10, p. 128-129]. The followmg example, Steinitz first, demonstrates that the summation range of a complex series may be a line in the complex plane. Let g be any line in the plane, a a point on g, and w =I= (0, 0) any point co on the line through the origin parallel to g. Let L 3n be any conditionally n=l convergent series of real numbers. Using standard vector addition it is easy to see that any number of the form a + rw, where r is a real number, will be a point 00 on the line g. By Riemann's theorem, the series L 3n can take on any real n=l value r. Hence the complex series a + a 1 w + ~ w + ··· can take on every complex value of the form a +rw. Thus the line g is the summation range of the complex senes. Steinitz second example demonstrates that a series of complex numbers can be rearranged to converge to any point in the complex plane. He begins 00 00 \\ith two conditionally convergent series of real numbers L 3n and L bn. 00 Again using Riemann's theorem, L 3n can be rearranged to converge to any n=l 00 real number a, and L bn can be rearranged to converge to any real number b. n=l Thus the series formed from the numbers a 1 , b 1 i, ~' b 2 i, ···can be rearranged .. to converge to any point in the complex plane. Steinitz makes two assertions in his introduction: first, that the summation range of a conditionally convergent series of complex numbers will always be n=l n=l
552 PI MU EPSILON JOURNAL either a line in the complex plane or the entire plane; second, that the summation range of a conditionally convergent series in n-dimensional space will always be a linear manifold or the whole space [10, p. 129]. Steinitz initial work was read by E. Landau who informed him of Levy's work and expressed some doubt about the correctness of Levy's results. In the introduction to his paper, Steinitz states that he arrived at his results in 1906 and was totally unaware of Levy's work [10, p.130]. This is obviously the case, as the reader will see, for his approach to determining the set of all sums of a conditionally convergent series is somewhat different from and more modem than Levy's. Levy's work, according to Steinitz, is brief, disconnected, and unclear. He says the work is incomplete, especially in the higher dimensional cases. He acknowledges that Levy must be given credit for proving the result for conditionally convergent series of complex numbers, but says that, although Levy stated the result for series in n-dimensional space, an actual proof requires morethanjustgeneralizinghis (Levy's) procedure [10, p. 130]. In section Vof his paper Steinitz gives his original proof for n-dimensional space and specifically indicates which points can be deduced from Levy's work and which cannot. In section VI of his paper Steinitz offers a proof using an entirely different approach, one which is shorter and more direct. In order to do this Steinitz develops an extensive and cumbersome amount of material on convex systems of rays and convex bodies. However, one gains some insight into his general approach when he discusses a series rearrangement theorem for a vector space of complex numbers with n units [ 10, p. 173]. Steinitz discusses in detail how certain conditionally convergent series of complex numbers may be rearranged to converge to any given complex number in the vector space. A slight modification ofhis approach leads to the following ideas for an. Let L xk be a conditionally convergent series of vectors in an with k=l the following property. Ifr ={xk};=I and r* is the set of all possible partial sums of elements from r, then r* is totally unbounded, i.e., unbounded in all 00 directions. Now let a be any vector in an. To obtain a rearrangement of L xk k=l which converges to a , Steinitz proceeds in the following manner. Define REARRANGEMENT OF SERIES, SMITH 553 gk = lubhk{llxJII}. Using the dyadic number system, Steinitz defines an ordering on the partial sums as follows. Given any partial sum II, replace each summand xk by 2k .. This detemtines a natural number for each partial sum, and thus also an orde"i·mg on the set of partial sums. Using results proven earlier in his paper, Steinitz shows that based on this orderu· m one can obtain a partial sum II which has x 1 as a summand and which satisfies the condition 1111 - a II - ngl . where n is the dimension of the space. He chooses the first partial sum II (using the ordering above) that satisfies this condition and calls it 11 1 • Now let r 1 denote the set of vectors which remain when the vectors occurring in 11 1 are removed from r. Then one can choose a partial sum II from r; which contains the first element from r 1 as a summand and satisfies the condition II III+ II- (J 11 - ng2. Steinitz chooses the first such partial sum (using the ordering above) and defines n2 =II1 + rr. Continuing in this fashion yields a sequence of partial sums Ilk-+ a and thus a rearrangement of L xk which has sum a. k=l The following example demonstrates Steinitz procedure. In a 2 , let r be the sequence (I.O),(O,l)( - 2 1 ,o) .( o, ~1 ) .( ~o) ( o.~) .( ~1 .o) .( o, ~1) ,. .. , and let a= (-1, 1). In this case, g 1 = 1, so ng 1 = 2. We want to fmd the first partial sum II such that 1111 - a II s 2. Taking the first vector from the sequence ~1elds 11(1, 0)- (-1, 1)11 = 11(2, -1)11 which is greater than 2, so another partial sum must be found. Consider the partial sum consisting of the first two vectors in r, 1 e., (1, 0) and (0, 1). Then 11(1, 0) + (0, I)- (-1, 1)11 = 11(2, 0)11 s 2. Since this IS the first partial sum (using Steinitz dyadic ordering scheme) that satisfies the condition 1111- all s 2, 11 1 =(I, 0) + (0, 1 ). Now r 1 is the set
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