SOME GENERALIZATIONS OF OLIVIER S THEOREM

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1 SOME GENERALIZATIONS OF OLIVIER S THEOREM Alai Faisat, Sait-Étiee, Georges Grekos, Sait-Étiee, Ladislav Mišík Ostrava (Received Jauary 27, 2006) Abstract. Let a be a coverget series of positive real umbers. L. Olivier proved that if the sequece (a ) is o-icreasig the lim a = 0. I the preset paper: (a) We formulate ad prove a ecessary ad sufficiet coditio for havig lim a = 0; Olivier s theorem is a cosequece of our Theorem 2.. (b) We prove properties aalogous to Olivier s property whe usual covergece is replaced by I-covergece, that is covergece accordig to a ideal I of subsets of N. Agai Olivier s theorem is a cosequece of our Theorem 3., whe oe takes as I the ideal of all fiite subsets of N. Keywords: coverget series, Olivier s theorem, ideal, I-covergece, I-mootoicity MSC 200: 40A05, 40A35, B05. Itroductio Durig last decades it was show that several rigid mathematical cocepts allow meaigful ad applicable extesios. I this ote we will deal with the cocept of covergece of sequeces with respect to a give ideal of subsets of positive itegers, so called I-covergece. The stadard covergece is a special case of such covergece with respect to the ideal of all fiite subsets of N = {, 2,... }. We will study the I-covergece variat of a classical result i mathematical aalysis, the Olivier s theorem. The well kow simple ecessary coditio for covergece of series states that the limit of the sequece of its terms is zero. I the case whe all terms of the series The research has bee supported by the Europea Regioal Developmet Fud i the IT4Iovatios Cetre of Excellece project (CZ..05/..00/ ), by the regioal grat RRC/04/200 DT ad the grat CAESAR ANR-2-BS0-00.

2 are positive ad their sequece is o-icreasig, some additioal iformatio about the speed of this covergece ca be deduced. I 827 L. Olivier [5] proved the followig theorem. Note that it is also kow as Abel - Prigsheim s Theorem (see e.g. [], Theorem 5.2.2). Theorem.. [5] Let (a ) N be a o-icreasig sequece of positive umbers such that the correspodig series a is coverget. The lim a = 0. This result was later geeralized ad exteded by several authors, see for example [2], [4], [6]. I this ote we will exted ad geeralize this result i two steps. First we prove a ecessary ad sufficiet coditio for the series of positive terms N a to fulfil lim a = 0. The we will exted this result i terms of I-covergece. Here we briefly metio some ecessary defiitios ad properties related to this cocept. A ideal I o N is ay oempty proper subclass of 2 N which is closed with respect to subsets ad fiite uios, i.e. A I, X A X I ad A I, B I A B I. Usually ideals are used to express that sets belogig to them are small i some sese. The class of all fiite subsets of N forms the ideal usualy deoted by I f. As further examples of ideals ca serve either the class I d of all subsets of N havig asymptotic desity 0, or the class I c of all subsets of N such that the sum of their reciprocals coverges. A ideal I is called admissible if it cotais all fiite subsets of N, i.e. I f I. Defiitio.. Let I be a admissible ideal ad (a ) be a sequece of real umbers. We say that (a ) is I-coverget to L if for every ε > 0 the set { N; a L ε} belogs to I. We deote this by I- lim a = L. Note that if this limit exists, the it is uique ad also that the stadard cocept of covergece of sequeces is exactly the I f -covergece. Also ote that for admissible ideals I J the implicatio (.) I lim a = L J lim a = L holds. For ay ideal I the dual filter is the class F(I) = {N \ A; A I}. Opposite to ideals, filters express that sets belogig to them are big i some sese. There is aother cocept of covergece with respect to a ideal referrig to its dual filter. 2

3 Defiitio.2. Let I be a admissible ideal ad (a ) a sequece of real umbers. We say that (a ) is I -coverget to L if there is a set K = {k < k 2 <... } i the dual filter F(I) such that lim a k = L. We deote this fact by I - lim a = L. Note that I -covergece is a stroger cocept tha I-covergece, i.e. I - lim a = L I- lim a = L, but for may ideals both cocepts coicide. These ideals are exactly those fulfillig the followig (AP ) property (see [3]). Defiitio.3. We say that a admissible ideal I satisfies the (AP) coditio if for every sequece of mutually disjoit sets (I ) from I there exists a sequece (J ) of sets J I such that (i) each set J I, =, 2,..., is fiite ad (ii) J I. For more iformatio o I-covergece we refer to [3]. 2. Stadard covergece There is a atural questio: how much ca the coditio of mootoicity of the sequece (a ) i Olivier s theorem be weakeed so that the coclusio lim a = 0, at least i some weaker sese, still holds? Of course, oe ca suppose that the mootoicity coditio holds except for a fiite umber of terms, but this represets a very cheap extesio of the origial result. Let a = (a ) be a real sequece with positive terms such that the correspodig series a coverges. Let S = partial sum, i.e. S = k= a ad, as usual, we deote by S the -th a k. For a give N let P = P (a) = {k {, 2,..., }; a k a }, R = R (a) = {k {, 2,..., }; a k < a } ad p = (a k a ), ad r = (a a k ). k P k R Notice that both p ad r are oegative. Thus, we have (2.) S = p r + a ad a r > 0. 3

4 Lemma 2.. For every coverget series a with positive terms we have (2.2) lim p = a = S. Proof. First, otice that, as p < S < S, the sequece (p ) is bouded by S. Thus, to prove (2.2), it is sufficiet to prove that for every ε > 0 there is a 0 N such that for all > 0 we have p > S 2ε. So, choose a ε > 0 ad let N be such that S > S ε. Let 0 > be a iteger such that a < ε ad a mi{a, a 2,..., a }, i.e. {, 2,..., } P, hold for all > 0. The we have for > 0 p = (a k a ) (a k a ) = S a > S ε ε k P k= which completes the proof. Usig (2.) ad (2.2) we deduce the followig theorem. Theorem 2.. For every coverget series a with positive terms we have (2.3) lim (a r ) = 0. A immediate cosequece of Theorem 2. is the followig corollary. Corollary 2.. Let a be a coverget series of positive terms. The (2.4) lim r = 0 lim a = 0 I particular (2.4) geeralizes the classical Olivier s theorem, i which r = 0 for every N. We will coclude this sectio with aother observatio based o Theorem 2.. For a sequece a = (a ) let us deote M = M(a) = { N; P = {, 2,..., }}, i.e. M if ad oly if a = mi{a, a 2,..., a }. Note that by the covergece of a we have that lim a = 0 ad, cosequetly the set M is ifiite. As r = 0 if ad oly if M, Theorem 2. immediately yields the followig propositio. 4

5 Propositio 2.. Let a = (a ) be a sequece of positive real umbers such that the correspodig series a coverges. The (2.5) lim m M(a) ma m = I-mootoicity ad Olivier s theorem I this sectio we are iterested i a geeralizatio of Olivier s theorem i terms of I-covergece. To do so, we eed some cocept of I-o-icreasig sequeces. 3.. I-mootoicity. Let I be a admissible ideal of subsets of N. Perhaps, the most atural way to defie the cocept of I-o-icreasig sequeces is the followig oe. Defiitio 3.. Let I be a admissible ideal of subsets of N ad let (a ) be a sequece of positive real umbers. We say that the sequece (a ) is I -o-icreasig (i symbols : (a ) I ) if there is a set K = {k < k 2 <... } i the dual filter F(I) such that the sequece (a k ) is o-icreasig. Here is also a stroger versio. Defiitio 3.2. Let I be a admissible ideal of subsets of N ad let a = (a ) be a sequece of positive real umbers. We say that the sequece a is I -o-icreasig (i symbols : a I ) if M(a) F(I). These star defiitios refer to the existece of a set i the dual filter, istead of assumig that all sets fulfillig a particular property belog to the origial ideal. Whe searchig for a o-star versio of defiitio, the followig oe ca be cosidered. Defiitio 3.3. Let I be a admissible ideal of subsets of N ad let a = (a ) be a sequece of positive real umbers. We say that the sequece a is I-o-icreasig (i symbols : a I ) if I- lim r = 0, i.e. if ad oly if for every ε > 0 the set T ε (a) = { N; r ε} belogs to I. It is atural to ask if there are some relatios amog the above defied cocepts. We prove hereo that the situatio is as the followig diagram shows. 5

6 P ROP.3. {I } P ROP.3.2 {I ad I } EX.3.7 {I } {I } EX.3.6 The oly geeral relatio amog the above cocepts is preseted i the followig propositio. Propositio 3.. Let a sequece of positive terms (a ) be I. The it is also both I ad I. The proof of the above statemet follows immediately from the correspodig defiitios ad is omitted. The ext propositio says that i o admissible ideal, except I f, the opposite implicatios hold. Propositio 3.2. Let a admissible ideal I cotai a ifiite set: I I f. The there exists a coverget series of positive umbers such that the correspodig sequece is both I ad I but is ot I. Proof. Let I = { = i < i 2 <... } be a ifiite set i I. We will costruct the required sequece a = (a ) by iductio. I the first step let us choose a positive umber a such that ad put (i 2 i )a < 2 a 2 = a 3 = = a i2 = 2a. Now choose a a i2 < 2 mi{a,..., a i2 } such that ad put (i 3 i 2 )a i2 < 2 2 a i2+ = a i2+2 = = a i3 = 2a i2. Cotiuig this process, i a geeral step we choose a < 2 mi{a,..., a i } such that a i (i + i )a i < 2 ad put a i + = a i +2 = = a i+ = 2a i. 6

7 For this sequece we have r = 0 if ad oly if I, thus (a ) is ot I. Deotig J = (N \ I) F(I), the subsequece (a ) J is o-icreasig, thus (a ) is I. To show that (a ) is also I choose (i k, i k+ ) ad calculate r = j,a >a j (a a j ) = a a ik = a ik < 2 k 0 as, so (r ) J teds to 0. Cosequetly, I- lim r = 0 ad a is I. Fially, the series a coverges : a = I a + J a < m= 2 m + m= = 3. 2m We coclude this subsectio with two examples providig the fact that I ad I are idepedet for geeral I. Notice that i a special case I = I f the implicatio I f I f holds. Example 3.. There is a admissible ideal I ad a real sequece a = (a ), a > 0, a <, which is I but ot I. Ideed, let I be a admissible ideal such that the set K = {2 k ; k N} belogs to its dual filter F(I). For k N let us defie {, if 2 k < < 2 k ; a = 2 k+, if = 2 k. 2 k The the subsequece (a ) K is o-icreasig, thus (a ) is I. O the other had, for each = 2 k K we have ( r 2 k = (a 2 k a j ) = 2 k ) 2 k+ = 2k 2 k+ 4 j<2 k, a j<a 2 k as k verifyig that (a ) is ot I. 2 k <j<2 k Example 3.2. There is a admissible ideal J ad a real sequece (b ) with b > 0, b <, ad such that (b ) is J but ot J. Let L = {2m; m N} ad L c = N \ L. Defie a admissible ideal J by J = {A B; A L c, B is fiite} =< I f, L c >, 7

8 that is J is the smallest admissible ideal cotaiig L c. Deote by F(J ) its dual filter ad otice that C F(J ) if ad oly if C cotais almost all members of L. The sequece b is defied as follows: Evidetly b > 0,, if L c ; 2 b =, if = 2m, m is odd; 2 2, if = 2m, m is eve. b <, b 0, so by Corollary 2. we have r 0 ad (b ) is J. Let us prove that (b ) is ot J. Suppose the cotrary, that is suppose that there is some T F(J ) such that (b ) T is o-icreasig. The T cotais almost all members of L, say T L, L \ L is fiite ad, by assumptio, (b ) L is o-icreasig. O the other had, for all odd m 3 we have a cotradictio. (2m) 2 > 2 < 2(m+) (2(m + 2)) 2, 3.2. Ideal variats of Olivier s Theorem. We have the followig I-Olivier theorem. Theorem 3.. Let a be a coverget series with positive terms ad I be a admissible ideal of subsets of N. The (a ) I I- lim a = 0. Proof. Sice lim (a r ) = 0 by Theorem 2., we have I- lim (a r ) = 0. The coditio (a ) I meas that I- lim r = 0; the result follows. Applyig Theorem 3. ad Propositio 3. to a coverget series with positive terms yields that if (a ) is I the I- lim a = 0. But we obtai eve more. Theorem 3.2. Let I be a admissible ideal of subsets of N ad let a = (a ) be a sequece of positive terms such that a is coverget. The (a )I I lim a = 0. 8

9 Proof. By hypothesis M(a) F(I); by Propositio 2. (3.) lim m M(a) ma m = 0; hece I lim a = 0. Now there is a questio : ca the assumptio that (a ) is I be relaxed to I? I other words, is the I -Olivier theorem true? The followig example shows that the aswer is, i geeral, o. We costruct a ideal I ad a I sequece (a ) with positive terms such that the correspodig series is coverget, but I - lim a = 0 fails. Example 3.3. Let S be the set of squares ad S c = N\S be the set of all positive itegers ot beig squares. Let I = {A B; A S c ad B is fiite} =< I f, S c >. Now defie the sequece (a ) as follows. { a =, if S c ; 2, if S. The a coverges, (a ) is I as the subsequece (a ) S is o-icreasig ad S F(I). We are goig to show that I - lim a = 0 fails. Suppose the cotrary, i.e. I - lim a = 0. The there is a set K = {k < k 2 <... } F(I) such that lim k a k = 0. O the other had, by defiitio of I, K cotais ifiitely may k s, for which k a k =, a cotradictio. We see from the above example that the star versio of Olivier s theorem does ot hold i geeral. Thus it would be iterestig to characterize those ideals for which this takes place. To that purpose, let us say that a ideal I satisfies the (*) coditio if for every coverget series a with positive terms, the implicatio (*) (a ) is I I - lim a = 0 holds. The ext theorem says that a ideal I satisfies (*) provided o set i I is too big. Before statig this theorem, let us recall that for a set J = {j < j 2 <... } N the lower ad upper asymptotic desities, respectively, are defied ad deoted by d(j) = lim if j ad d(j) = lim sup j. 9

10 Theorem 3.3. Let I be a ideal such that d(j) < for every J I. The I satisfies (*). Proof. Assumig (a ) is I, we ca choose K = {k < k 2 <... } F(I) such that (a k ) is o-icreasig. Of course, <, thus Olivier s theorem a k ca be applied to the sequece (a k ) to get lim a k = 0. By assumptios, α = lim if k = d(k) = d(n \ K) > 0. Thus we have lim sup k lim a k = α 0 = 0, which implies lim k a k = 0, so that I lim a = Applicatio Lemma 2. proves a ice asymptotic behavior of p. O the other had, both statemet (2.4) ad Theorem 3. say that it is ot the case of the asymptotic behavior of r. To see this, it is sufficiet to cosider a sequece a = (a ), a > 0, such that the series a is coverget ad a does ot ted to 0. Nevertheless, some iformatio o the asymptotic behavior of r ca be derived. The iequality i (2.) implies a > r ad, cosequetly, we obtai (4.) r < S <. Recall (Defiitio 3.3) that, for every ε > 0, T ε (a) = { N; r ε}. The (4.) implies r S > r ε ad, cosequetly, T ε (a) I c = series with positive terms, we have T ε (a) { X N; x X (4.2) I c - lim r = 0. T ε (a) Sice I c - lim (a r ) = 0, the followig theorem [6] holds. x < }. Thus, for every coverget Theorem 4.. (Šalát, Toma) For every coverget series a of positive terms ad each ideal I I c we have I- lim a = 0. 0

11 Remark 4.. It is a easy exercise to show that the ideal I c satisfies the (AP) coditio ad, aalogously to (.), for ay pair of admissible ideals I J the implicatio I - lim a = L J - lim a = L holds for every sequece (a ). Cosequetly, i the previous theorem the I- covergece ca be substituted by the I -covergece. Theorem 4.2. For every coverget series a of positive terms ad each ideal I I c we have I - lim a = 0. Refereces [] S. Badyopadhyay: Mathematical Aalysis - Problems ad Solutios, Academic Publishers (2006), ISBN [2] K. Kopp: Theorie ud Awedug der uedliche Reihe. 3 Aufl., Spriger 93 [3] P. Kostyrko, T., Šalát, W. Wilczyski: I-covergece. Real Aalysis Exchage 26 (2000/0), o. 2, [4] J. Krzyž: Twierdzeie Oliviera i jego uogóleia. Prace Mat. 2 (956), [5] L. Olivier: Remarques sur les séries ifiies et leur covergece. J.Reie Agew. Math. 2 (827), 3 44 [6] T. Šalát, V. Toma: Olivier s theorem ad statistical covergece. Aales Mathématiques Blaise Pascal 0 (2003), Authors addresses: Alai Faisat, Départemet de Mathématiques ad Istitut Camille Jorda, Uiversité Jea Moet Sait-Étiee cedex 2, Frace (Sait-Étiee), 23 rue du Dr Paul Michelo, faisat@uiv-st-etiee.fr, Georges Grekos, Départemet de Mathématiques ad Istitut Camille Jorda, Uiversité Jea Moet (Sait-Étiee), 23 rue du Dr Paul Michelo, Sait-Étiee cedex 2, Frace grekos@uiv-st-etiee.fr, Ladislav Mišík, Departmet of Mathematics ad IT4 Iovatios Cetre of Excellece, Uiversity of Ostrava, 30. duba 22, Ostrava, Czech Republic ladislav.misik@osu.cz.