ON SOME EQUIVALENCE RELATION ρ IN THE CLASS OF BOUNDED SEQUENCES OF POSITIVE REAL NUMBERS

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1 International Journal of Pure and Applied Mathematics Volume 94 No , ISSN: (printed version); ISSN: (on-line version) url: doi: PAijpam.eu ON SOME EQUIVALENCE RELATION ρ IN THE CLASS OF BOUNDED SEQUENCES OF POSITIVE REAL NUMBERS Micha l Różańsi 1, Damian S lota 2 Marcin Szweda 3, Roman Witu la 4 1,2,3,4 Institute of Mathematics Silesian University of Technology Kaszubsa 23, Gliwice, POLAND Abstract: In this paper some equivalence relation in the class of bounded sequences of positive numbers is discussed. This relation enables to understand better the permutations preserving divergence of the same rate, introduced by Johnston. It also gives the possibility to prove, in ew way, the combinatoric characterization of these permutations. AMS Subject Classification: 40A05, 05A05 Key Words: ivergence of series at the same rate, permutation preserving divergence of the same rate 1. Introduction Motivation for preparing this paper was Johnston s paper [3] and the rearrangements that preserve rates of divergence discussed in it. Let us recall, after Johnston, some notation. Received: May 4, 2014 Correspondence author c 2014 Academic Publications, Ltd. url:

2 252 M. Różańsi, D. S lota, M. Szweda, R. Witu la Definition 1. Let a and b bethe divergent series of positive terms. We say that a and b diverge at the same rate if 0 < α = liminf n a limsup n b a = β <. (1) b If α = β = 1 in the above relation, then we say that these series are asymptotic and we write a b. Definition 2. A permutation p on N is called the permutation preserving divergence of the same rate if for each divergent series a, with positive and bounded sequence of terms, both series a and its p-rearrangement a p() diverge at the same rate. Johnston observed that the family of all permutations on N preserving divergence of the same rate forms a group and it is not the normal subgroup of the group of all permutations on N. Furthermore, he proved the following combinatoric characterization of these permutations. Theorem 3. Let p be a permutation on N. Then p is a permutation preserving divergence of the same rate if and only if there exists N = N(p) N such that card(i \p(i)) N (2) for every finite interval I N. In this paper we introduce some equivalence relation ρ, defined in the family of bounded sequences of positive numbers, which gives us the possibility to investigate more carefully the permutations preserving divergence of the same rate, presented and discussed by E. Johnston. This relation enables also to obtain easily the Johnston characterization (see Theorem 3 from above) shedding some new light on condition (2) (see Corollary 10 at the end of this paper). 2. Equivalence Relation ρ Let d denote the class of bounded sequences of positive real numbers. We introduce in class d relation ρ defined in the following way: { }ρ{ } for every increasing sequence {n i } of natural

3 ON SOME EQUIVALENCE RELATION ρ IN numbers both series i and i are simultaneously either convergent or divergent, for every { },{ } d. One can easily verify that ρ is the equivalence relation in d. Let { },{ } d. If { ani = + and bni < + for some increasing sequence {n i } N, (3) then sequences { } and { } belong certainly to different equivalence classes with respect to relation ρ and vice versa. It appears that condition (3) is equivalent with the existence of increasing sequence {n i } N such that ani = + and lim i i = + (see Theorem 4). It can be also shown (see Theorem 6) that if sequences { },{ } d belong to the same equivalence class with respect to relation ρ, then there exist the constants m,m > 0, m M, such that >M < + and <m < +, (if the set of subscripts is empty then we tae the sum as equal to 0), from which we get easily that 0 < liminf and limsup < + (4) (see Theorem 7). We note that the above characterization is not sufficient for belongingness of sequences { } and { } from d to the same equivalence class with respect to relation ρ (see Remar 9). Whereas if { },{ } d and liminf = 0 or limsup = +, then from (4) it follows that sequences { } and { } belong to different equivalence classes with respect to relation ρ. We present now the theorems mentioned above.

4 254 M. Różańsi, D. S lota, M. Szweda, R. Witu la Theorem 4. Let{ },{ } d. For existenceofanincreasingsequenceof positive integers {n i }, such that i = + and i < +, it is necessary and sufficient that there exists an increasing sequence {n i } N such that ani = + and lim an i i = +. Proof. Let c n,d n > 0,n 1. We show that if c n = + and d n < + then there exists an increasing sequence {n i } N such that cni = + and lim c n i d ni = +. (5) First of all, let us notice that then, for each M > 0, if {n i } = {n N c n Md n } then cni = +. By this property we define now two auxiliary sets of indices {t n } and { n }. Let In general N 1 = {n N c n d n }, 1 = minn 1, { } t 1 = min t N 1 c n 1. 1 n t, n N 1 N s = {n N n > t s 1 i c n sd n }, s = minn s, { } t s = min t N s c n 1, for every s N, s > 1. s n t, n N s One can easily verify that the increasing sequence {n i } of all elements of the following set {n N there exists s N such that s n t s and n N s } fulfils conditions (5). Now, let us assume that there exists an increasing sequence {n i } N such that i = + and lim an i i = + and i,i 1, i 1. Let s = min{ N i 2 s i for every i }, s 1. We form two auxiliary sequences {t s } and {l s }: l 1 = 1, { t 1 = min t N t l 1 and t } i 1, i=l 1

5 ON SOME EQUIVALENCE RELATION ρ IN l s = max{t s 1 +1, s }, { t s = min t N t l s and t } i 1, i=l s for every s N, s > 1. Let {n i } i=1 be the increasing sequence of all elements of set {n i l s i t s }. Then we have bn an s N = i i s 1 i=l s s 1 t s = t s i 1 = +, i s 1 i=l s s 1 ( 1 ts ) 2 s i i=l s s s = 2 < +. Corollary 5. Let, > 0, lim = 0. If = +, then there exists an increasing sequence {n i } N such that ani = + and ani i < +. Proof. Itisenoughtonoticethat = + andlim an = lim 1 = +, and to apply Theorem 4. Theorem 6. Let { },{ } d, = = +. If sequences { } and { } belong to the same equivalence class with respect to relation ρ, then there exists M > 0 such that < + (if {n N > M } =, >M then := 0). >M Proof. Let us suppose that for each M > 0 >M = +. (6) For simplicity we assume that 1, n 1. We form two auxiliary sequences { s } and {t s } of positive integers 1 = min{ N a > b },

6 256 M. Różańsi, D. S lota, M. Szweda, R. Witu la { t 1 = min t N t 1 and } 1, 1 n t s = min{ N > t s 1 and a > 2 s 1 b }, { } t s = min t N t s and 1, s n t >2 s 1 for every s > 1. Let {n i } be the increasing sequence of all elements of set {n N s n t s and > 2 s 1 }. s N We can easily chec that i = + and i < +. Indeed, we have ani = s 1 s n t s >2 s 1 s 11 = +, bni = s 1 s n t s >2 s 1 s s 1 s n t s >2 s 1 s 1 2 = 4 < +. 2s 1 Thus the sequences { } and { } belong to the different equivalence classes with respect to relation ρ which contradicts the assumption. Theorem 7. Let { },{ } d. If sequences { } and { } belong to the same equivalence class with respect to relation ρ, then 0 < liminf limsup < +. Proof. It is sufficient to confine the considerations to the case = bn = +. Let M > 0 be such as in Theorem 6. Let us put 0 if {n N 1 n and > M } =, := otherwise, 1 n >M

7 ON SOME EQUIVALENCE RELATION ρ IN and similarly 0 if {n N 1 n and > M } =, := otherwise. 1 n >M Since n 1 < +, therefore = + must hold and from this, for M sufficiently large N we get = + 1 n M + 1 n M + 1 n M 1 n M M + n 1 1 n M But condition = + implies that also = +, thus M M ( )/( lim ) = 0, which gives the boundedness of sequence 1 n M {( )/( )} n. Analogically we prove that sequence b. is bounded as well. {( )/( )} Corollary 8. Let { },{ } d. If the one of the following conditions hold lim inf = 0 or limsup = +, then the sequences { } and { } belong to the different equivalence classes with respect to relation ρ. ( )/( ) In other words, if limsup = +, then there exist the positive integers {n i } such that ani = + and bni < +.

8 258 M. Różańsi, D. S lota, M. Szweda, R. Witu la Remar 9. Theorem 7 cannot be inverted. More precisely, there exist sequences { },{ } d satisfying condition (1) and belonging to the different equivalence classes with respect to relation ρ. For this purpose let us set for every n N. = n 1, { 1 if n = 2, N, = 2 n if n is odd, ( )/( ) We easily chec that lim = 1 and, in spite of this, sequences { } and { } belong to the different equivalence classes with respect to relation ρ. Indeed, we have 2 1 = 1a = +, = 1b < +. 1 First we observe that if 3. Proof of Johnston s Theorem 3 sup{card(i \p(i))} =, where the supremum is taen over all finite intervals I in N, then there exists a sequence {I n } of finite intervals in N such that card(i n ) as n, I n p(i n ) =, and < l for any I n p(i n ), l I n+1 p(i n+1 ) and n N. Hence we get A p(a) =, where A := I n, and the set A is infinite. This implies that there exists a n N series a of positive terms such that a is divergent and simultaneously A

9 ON SOME EQUIVALENCE RELATION ρ IN the series a p() is convergent, which means that p is not a permutation on A N preserving divergence of the same rate. On the other hand, if M N and card(i \ p(i)) M, for every finite interval I in N, then for every n N there exist the disjoint sets A n,b n N such that p([1,n]) = ([1,n]\A n ) B n, card(a n B n ) = M. Next, if {a } is a bounded sequence of positive terms and the series a is divergent then a as n, a p() = a A n a + B n a, which implies that ( )/( ) n a p() a 1 and it means that p is the permutation preserving divergence of the same rate. Corollary 10. Let p be a permutation on N. If sup{card(i \p(i))} =, (7) where supremum is taen over all finite intervals I in N, then there exists a divergent series with { } d such that the sequences { } and {a p(n) } belong to the different equivalence classes with respect to relation ρ. We note that condition (7) could be replaced by the following strange condition: there exists an infinite set A N such that A p(a) =. Furthermore, if for every { } d the sequences { } and {a p(n) } belong to the same equivalence class with respect to relation ρ, then there exist N = N(p) N such that p(n) = n for every n N, n N.

10 260 M. Różańsi, D. S lota, M. Szweda, R. Witu la 4. Final Comments The subject matter using or referring to the quotients of the nth partial sums of series, discussed here, appears in many papers of various authors (see [1], [2], [4]). With no doubts, this subject represents a very useful technical tool and we thin that the results presented in this paper can contribute in consolidating this matter. Next, from Theorem 3 it follows that every permutation p on N preserving divergence of the same rate is simultaneously a convergent permutation, i.e. permutation satisfying the following condition: for every convergent real series an (either complex series or even vector series in complete normed space) the p-rearranged series a p(n) is also convergent. It is well now that the convergent permutations preserve also the sum of rearranged series (see [5], [6], [7]). We note that if q is a bijection of N onto some subset W of N with finite complement and the condition (2) holds, then q is a bijection preserving convergence in sense of definition of the convergent permutation p given above. But there is one spectacular difference between these, so called, convergent bijections and convergent permutations. Convergent bijections generally do not preserve the sum of rearranged series. References [1] E. Hensz-Ch adzyńsa, R. Jajte, A. Pasziewicz, Random stain, Probab. Math. Statist., 18, No. 1 (1988), [2] D. Holy, L. Matejica, L. Pinda, Some remars of faster convergent infinite series, Math. Slovaca, 62, No. 4 (2012), , doi: /s [3] E.H. Johnston, Rearrangements that preserve rates of divergence, Can. J. Math., 34, No. 4 (1982), , doi: /CJM [4] F. Prus-Wiśniowsi, On inclusion between Watermann classes and Chanturyia classes, Tatra Mt. Math. Publ., 19 (2000), [5] R. Witu la, Convergence preserving functions, Nieuw Arch. Wis. IV. Ser., 13, No. 1 (1995), [6] R. Witu la, Permutations preserving the sum of rearranged real series, Cent. Eur. J. Math., 11, No. 5 (2013), , doi: /s x.

11 ON SOME EQUIVALENCE RELATION ρ IN [7] R. Witu la, Permutations preserving the convergence or the sum of series a survey, in R. Witu la, D. S lota, W. Ho lubowsi (eds.), Monograph on the Occasion of 100 th Birthday Anniversary of Zygmunt Zahorsi, Wyd. Pol. Śl., Poland (2014), in press.

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