ANNALES MATHÉMATIQUES BLAISE PASCAL. Tibor Šalát, Vladimír Toma A Classical Olivier s Theorem and Statistical Convergence

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1 ANNALES MATHÉMATIQUES BLAISE PASCAL Tibor Šalát, Vladimír Toma A Classical Olivier s Theorem and Statistical Convergence Volume 10, n o 2 (2003), <htt://amb.cedram.org/item?id=ambp_ _2_305_0> Annales mathématiques Blaise Pascal, 2003, tous droits réservés. L accès aux articles de la revue «Annales mathématiques Blaise Pascal» (htt://amb.cedram.org/), imlique l accord avec les conditions générales d utilisation (htt://amb.cedram.org/legal/). Toute utilisation commerciale ou imression systématique est constitutive d une infraction énale. Toute coie ou imression de ce fichier doit contenir la résente mention de coyright. Publication éditée ar le laboratoire de mathématiques de l université Blaise-Pascal, UMR 6620 du CNRS Clermont-Ferrand France cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques htt://

2 Annales Mathematiques Blaise Pascal 10, (2003) A Classical Olivier s Theorem and Statistical Convergence Tibor Šalát Vladimír Toma Abstract L. Olivier roved in 1827 the classical result about the seed of convergence to zero of the terms of a convergent series with ositive and decreasing terms. We rove that this result remains true if we omit the monotonicity of the terms of the series when the limit oeration is relaced by the statistical limit, or some generalizations of this concet. Résumé. L. Olivier démontrait en 1827 un résultat classique sur la vitesse de convergence vers zéro d une série convergente à termes ositifs décroissants. Nous démontrons que ce résultat reste valable si nous omettons la monotonie des termes de la série, en remlaçant l oération limite ar la limite statistique ou encore ar des généralisations de ce concet. 1 Introduction The above mentioned result of L. Olivier was ublished in [7]. 39 (see also [5]. 125) and claims that if a n a n+1 > 0, (n = 1, 2,... ) and a n < + then lim na n = 0. Simle examles show that without the monotonicity condition a n a n+1, (n = 1, 2,... ), the sequence (na n ) n 1 need not converge to zero. Examle 1. Let a n = 1 if n is a square i.e. n = n k2, (k = 1, 2,... ), and a n = 1 otherwise. Then a n 2 n > 0 (n = 1, 2,... ), a n < +, but lim na n 0 since k 2 a k 2 = 1, (k = 1, 2,... ). 305

3 T. Šalát and V. Toma Remark: As the unknown referee ointed out the examle can be strengthened taking a n = log n if n = k 2, in which case the sequence (na n n ) n 1 is not bounded. The notion which allows us to describe the behavior of the sequence (na n ) n 1 is the notion of statistical convergence introduced in aer [2], (see also [3], [10], and [9]) Definition: We say that a sequence (x n ) n 1 (of real or comlex numbers) statistically converges to a number L, and we write lim-stat x n = L, if for each ε > 0 the set A(ε) := {n : x n L ε} has zero asymtotic density, i.e. the limit 1 n d(a(ε)) := lim χ A(ε) (k) n exists and is equal to zero. Here χ A is the characteristic function of a set A. In what follows we will show that the sequence (na n ) n 1 statistically converges to 0 if a n < +, (a n > 0, n = 1, 2,... ) without the monotonicity assumtion on the sequence (a n ) n 1. The notion of statistical convergence was generalized using the concet of an admissible ideal I of subsets of ositive integers N = {1, 2,... }, that is I P(N) and I is additive (i.e. A, B I A B I), hereditary (i.e. B A I B I), containing all singletons and not containing N. Definition: (See [6].) We say that a sequence (x n ) n 1 I-converges to a number L and we write I-lim x n = L, if for each ε > 0 the set A(ε) := {n : x n L ε} belongs to the ideal I. An admissible ideal is for examle: I f := {A N : A is finite set}. Let us note that I f -lim x n = L means the same as lim x n = L. Some admissible ideals can be obtained using various concets of density of sets A N. Using the asymtotic density defined above we obtain the ideal I d := {A N : d(a) = 0}. Obviously I d -convergence means the statistical convergence. Another tye of density is the logarithmic density defined by means of lower and uer 306

4 Olivier s theorem and statistical convergence logarithmic density of a set A N: δ(a) := lim inf n χ A(k) 1 k, δ(a) := lim su log n n χ A(k) 1 k. log n If δ(a) = δ(a) =: δ(a) the number δ(a) is called the logarithmic density of the set A. Using the logarithmic density we can define the ideal I δ := {A N : δ(a) = 0}. A little bit more comlicated is the concet of the uniform density. For any A N, t 0, s 1 denote by A(t + 1, t + s) the number of elements of the set A [t + 1, t + s]. Put Then there exist := lim inf A(t + 1, t + s), t βs := lim su A(t + 1, t + s). u(a) := lim s s, t u(a) := lim s s called the lower and uer uniform density of A, resectively. We rove the β existence of lim s s only, since the roof for lim s s βs is similar. Let us s choose a fixed N. Then for any s N there exists t s 1 such that = A(t s + 1, t s + s) and simultaneously β A(t + 1, t + ) for every t t s. For any s N there exist unique integer numbers k s, r s 0 such that s = k s + r s with 0 r s 1. Then we have (with the convention A(x, y) = 0 if y < x) s = 1 A(t s + 1, t s + k s + r s ) = k s + r s 1 k s = [ A(t s (i 1), t s + i) + A(t s + k s + 1, t s + k s + r s )] k s + r s i=1 Hence βs ksβ s have lim ks u(a) su 1 k s+r s ksβ k s+r s and when s then also k s and so for fixed we = β. Therefore u(a) = lim inf β. Since obviously lim su there exists the limit lim s s s = su 1 β 307 s s s β and consequently β su, we can conclude that 1. If u(a) = u(a) =: u(a) then the

5 T. Šalát and V. Toma common value u(a) is called the uniform density of A (cf.[1]). Using the uniform density we can define the admissible ideal I u := {A N : u(a) = 0}. To comare the above defined ideals let us remember the lower and uer asymtotic density defined by d(a) := lim inf 1 n n χ A (k), d(a) := lim su 1 n n χ A (k). The following relations between these densities can be verified (cf.[1],[4]): 0 u(a) d(a) δ(a) δ(a) d(a) u(a) 1. Consequently we get the chain of inclusions for the above defined ideals: I f I u I d I δ. (1.1) The ideal which will lay an imortant role in the main theorem is the following one I c := {A N : a 1 < + }. It is well known (see [8]) that a 1 < + imlies d(a) = 0. So the a A following inclusion holds I c I d. (1.2) a A 2 Main Results The above mentioned Olivier s result can be formulated in the terms of I f - convergence as follows. If a n > 0 (n = 1, 2,... ), a n < + and a 1 a 2 a n a n

6 Olivier s theorem and statistical convergence then I f -lim na n = 0. In the sequel we are going to study the ideals I with the following roerty: If a n > 0 (n = 1, 2,... ), a n < + then I-lim na n = 0. (2.1) From Examle 1 we can conclude that the ideal I f does not have the roerty 2.1. Let us denote by S(T ) the class of all admissible ideals I, with the roerty 2.1. So we have that I f / S(T ). The following theorem claims a more useful fact. Theorem 2.1: Ideal I c is an element of S(T ). Proof: We roceed by contradiction. Let I c := {A N : a 1 < + } / S(T ). a A Then there exist numbers a n > 0 (n = 1, 2,... ) with a n < + such that the equality I c -lim na n = 0 does not hold. This means that there exists ε 0 > 0 for which A(ε 0 ) = {n : na n ε 0 } / I c. Hence from the definition of the ideal I c we get n 1 = +. For n A(ε 0 ) we have na n ε 0 and n A(ε 0) so a n ε 0 n for every n A(ε 0). Using this and the comarison criterion for infinite series we get a n ε 0 n 1 = +. n A(ε 0) n A(ε 0) So it must be also a n = + and this is a contradiction. The claim in the following lemma is a trivial fact about reservation of the limit. Lemma 2.2: Let I 1, I 2 be admissible ideals such that I 1 I 2. If I 1 - lim x n = L, then also I 2 -limx n = L. An obvious consequence of Lemma 2.2 is the following theorem. 309

7 T. Šalát and V. Toma Theorem 2.3: If I 1 I 2 are two admissible ideals and I 1 S(T ) then I 2 S(T ). Corollary 2.4: If I is an admissible ideal and I I c then I S(T ). Theorem 2.5: The ideal I c is the smallest element in the class S(T ) artially ordered by inclusion. Proof: Let I S(T ). We rove that for any set M = {m 1 < m 2 <... } I c we have M I. We can suose that M is an infinite set because if it were finite it belongs to I as it is an admissible ideal, hence contains all finite sets. Since M I c following the definition of the ideal I c we have 1 m k < + Let us define numbers a n (n = 1, 2,... ) as follows a mk = 1 m k (k = 1, 2,... ), a n = 1 n 2 + n for n N \ M. Then obviously a n > 0 (n = 1, 2,... ) and a n < + by the definition of numbers a n. Since I S(T ) we have I-lim na n = 0. This imlies that for each ε > 0 we have secifically M = A(1) I. A(ε) = {n : na n ε} I, The roblem of characterization of the class of all ideals having the roerty 2.1 was formulated orally by the second author in a discussion in the Seminar on real functions theory in Bratislava. The above results allow us to give such a characterization. Theorem 2.6: The class S(T ) consists of all admissible ideals I P(N) such that I I c. 310

8 Olivier s theorem and statistical convergence Proof: 1) If I S(T ) then after Theorem 2.5 we have I I c. 2) Let I be an admissible ideal and I c I. Due to Theorem 2.1 we have I c S(T ) and the Corollary 2.4 yields I S(T ). Remark: Referring to inclusions 1.1 and 1.2 we can claim that S(T ) contains as elements the ideals I d and I δ. Since the I d -convergence is in fact the statistical convergence we get a n < +, (a n > 0) = lim-stat na n = 0. In connection with Theorem 2.6 it is imortant to know how many ideals have the roerty 2.1 and how many don t have this roerty. We know yet that the ideal I f which yields usual convergence is not an element of S(T ). We give some more such examles. Examle 2. For every integer j 0 let D j := {2 j (2n + 1) : n = 0, 1,... }; then obviously D j = N. Let I be the set of all A N which intersect j=0 only finite number of the sets D 0, D 1,.... Then I is obviously an ideal which does not contain the set A = {1, 2, 2 2,..., 2 j,... } since A meets every D j but A is obviously an element of I c. Hence I c I. Examle 3. The ideal I u := {A N : u(a) = 0} where u(a) is the uniform density of the set A, is not an element of the class S(T ). To rove this let us choose A = A k with A k = {2 2k + 1,..., 2 2k + 2 k }, (k = 1, 2,... ). Obviously A I c. To rove that A / I u let us determine the uer uniform density of the set A (see [1]). To this end, consider the sets A [t + 1, t + s] with t 0, s N. If we take t k = 2 2k, s k = 2 k, (k = 1, 2,... ), then A(t k + 1, t k + s k ) = s k, (k = 1, 2,... ) and β (sk) := lim su A(t + 1, t + s k ) = s k and t β consequently u(a) = lim (s k ) k s k = 1. So we have roved that A / I u and hence I c I u The next two roositions give an idea of how many admissible ideals are aroriate for the analog of Olivier s theorem and how many can not be used to obtain this analog. 311

9 T. Šalát and V. Toma Proosition 2.7: There are infinitely many admissible ideals which are not elements of the class S(T ). Proof: For any infinite set M N we enlarge the ideal I f by adjunction of the set M defining a new ideal I M := {A B : A I f, B M}. If we choose M such that N\M is also infinite then I M is an admissible ideal. We can choose moreover M such that m M m 1 < + and then I M I c. Alying Theorem 2.5 we conclude that I M is not an element of S(T ). To see that there are infinitely many ideals of the tye I M it is sufficient to observe that I M I M if and only if the symmetric difference M M is an infinite set. Remark: P. Kostyrko observed that for each q, 0 < q < 1, the admissible ideal I (q) c := {A N : a A a q < + } is a roer subset of the ideal I c. So we get again by Theorem 2.5 infinitely many admissible ideals that do not belong to S(T ). Proosition 2.8: There are infinitely many admissible ideals which are elements of the class S(T ). Proof: Let us take a set M with m 1 = + such that N \ M is infinite. Then the ideal m M I M = {A B : A I c, B M} is an admissible ideal such that I c I M and consequently I M S(T ). To see that there are infinitely many such ideals I M let us write M = {m 1 < m 2 <... } with = +. Then by comarison test we have also m 1 k m 1 2k 1 = + = m 1 2k. If we take M = {m 2k 1 : k = 1, 2... } then I c I M I M and in this way we construct a decreasing chain of infinitely many admissible ideals of the class S(T ). References [1] T. C. Brown and A. R. Freedman. The uniform density of sets of integers and Fermat s last theorem. C. R. Math. Re. Acad. Sci. Canada, XII:1 6,

10 Olivier s theorem and statistical convergence [2] H. Fast. Sur la convergence statistique. Coll. Math., 2: , [3] J. A. Fridy. On statistical convergence. Analysis, 5: , [4] H. Halberstam and K. F. Roth. Sequences I. Oxford University Press, Oxford, [5] K. Kno. Theorie und Anwendung der unendlichen Reihen 3. Aufl [6] P. Kostyrko, T. Šalát, and W. Wilczński. I-convergence. Real Anal. Exch., 26: , [7] L. Olivier. Remarques sur les séries infinies et leur convergence. J. reine angew. Math., 2:31 44, [8] B. J. Powel and T. Šalát. Convergence of subseries of the harmonic series and asymtotic densities of sets of ositive integers. Publ. Inst. Math.(Beograd), 50(64):60 70, [9] I. J. Schoenberg. The integrability of certain functions and related summability methods. Amer. Math. Monthly, 66: , [10] T. Šalát. On statistically convergent sequences of real numbers. Math. Slovaca, 30: , Tibor Šalát Vladimír Toma Comenius University Comenius University Deartment of Mathematics Deartment of Mathematics Mlynská dolina Mlynská dolina Bratislava Bratislava SLOVAK REPUBLIC SLOVAK REPUBLIC toma@fmh.uniba.sk 313