In 1827 L. Olivier proved result about the speed of convergence to zero of the terms of convergent positive series with non-increasing terms so-called Olivier\u27s Theorem. T. Šalát and V. Toma made remark that the monotonicity condition in Olivier\u27s Theorem can be dropped if the convergence of the sequence (nan) is weakened by means of the notion of I-convergence for an appropriate ideal I. Results of this type are called a modified Olivier\u27s Theorem.

In connection with this we will study the properties of summable ideals Ih where h: R+→R+ is a function such that Σn∈Nh(n)=+∞ and Ih={A⊊N : Σn∈Ah(n)<+∞}. We show that Ih-convergence and Ih*-convergence are equivalent. What does not valid in general.
Further we also show that the modified Olivier\u27s Theorem is not valid for summable ideals Ih in generally. We find sufficient conditions for real function h: R+→R+ such that modified Olivier\u27s Theorem remains valid for ideal Ih

Baláž, Vladimír

Maťašovský, Alexander

Visnyai, Tomáš

English

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MATHEMATICAL COMMUNICATIONS 1Math. Commun. 28(2023), 1–9Ih-convergence and convergence of positive series∗Vladiḿır Baláž, Alexander Maťašovský†and Tomáš VisnyaiFaculty of Chemical and Food Technology, Slovak University of Technology in Bratislava,Radlinského 9, 812 37 Bratislava, SlovakiaReceived April 28, 2022; accepted October 4, 2022Abstract. In 1827, L. Olivier proved a result about the speed of convergence to zero ofthe terms of convergent positive series with nonincreasing terms, the so-called Olivier’stheorem (see [17]). T. Šalát and V.Toma in [20] made the remark that the monotonicitycondition in Olivier’s theorem can be dropped if the convergence of the sequence (nan) isweakened by means of the notion of I-convergence for an appropriate ideal I. Results ofthis type are called a modified Olivier’s theorem.In connection with this, we will study the properties of summable ideals Ih, where h : R+ →R+ is a function such that∑n∈N h(n) = +∞ and Ih ={A ( N :∑n∈A h(n) < +∞}.We show that Ih-convergence and Ih∗-convergence are equivalent. This is not valid ingeneral.Further, we also show that a modified Olivier’s theorem is not valid for summable idealsIh in general. We find sufficient conditions for a real function h : R+ → R+ such that amodified Olivier’s theorem remains valid for the ideal Ih.AMS subject classifications: 40A05, 40A35Keywords: I-convergence, convergence of positive series, Olivier’s theorem, admissibleideals1. IntroductionWe recall the basic definitions and connections that will be used throughout thispaper. Let N be the set of all positive integers, N0 = N ∪ {0}, and R+ the set of allpositive real numbers. A system I, ∅ ̸= I ⊆ 2N is called an ideal, provided that I isadditive (A,B ∈ I implies A∪B ∈ I) and hereditary (A ∈ I, B ⊂ A implies B ∈ I).The ideal is called nontrivial if I ̸= 2N. If I is a nontrivial ideal, then I is calledadmissible if it contains the singletons ({n} ∈ I for every n ∈ N). The fundamentalnotation shall be used is I-convergence introduced in the paper [14] (see also [5],where I-convergence is defined by means of the dual notion to the ideal so-calledfilter). The notion of I-convergence corresponds to the natural generalization of thenotion of statistical convergence (see [8, 19]).∗V.B. wishes to thank The Slovak Research and Development Agency (research project VEGA No.2/0119/23) for financial support. A.M. wishes to thank The Slovak Research and DevelopmentAgency (research project VEGA No. 1/0386/21) for financial support.†Corresponding author. Email addresses: vladimir.balaz@stuba.sk (V.Baláž),alexander.matasovsky@stuba.sk (A.Maťašovský), tomas.visnyai@stuba.sk (T.Visnyai)http://www.mathos.hr/mc c⃝2023 Department of Mathematics, University of Osijek2 V. Baláž, A. Maťašovský and T. VisnyaiDefinition 1. Let (xn) be a sequence of real (complex) numbers. We say that thesequence I-converges to a number L, and write I − limxn = L, if for each ε > 0 theset Aε = {n : |xn − L| ≥ ε} belongs to the ideal I.In what follows, we assume that I is an admissible ideal. Then for every sequence(xn) we immediately have that limn→∞ xn = L (classic limit) implies that (xn) alsoI-converges to a number L, but the opposite is not true. In other words, for anadmissible ideal I we have Ifin ⊆ I, where Ifin is the ideal of all finite subsets ofN and Ifin-convergence coincides with the usual convergence.Let Id = {A ⊆ N : d(A) = 0}, where d(A) is the asymptotic density of A ⊆ N(d(A) = limn→∞#{a≤n : a∈A}n , where #M denotes the cardinality of the set M).The usual Id-convergence is called statistical convergence. For 0 < q ≤ 1, theideal I(q)c ={A ⊂ N :∑a∈A a−q < ∞}is an admissible ideal. The ideal I(1)c ={A ⊂ N :∑a∈A1a < ∞}is usually denoted by Ic.I-convergence satisfies usual axioms of convergence i.e., the uniqueness of thelimit, the arithmetical properties, etc. The class of all I-convergent sequences is alinear space (see [14]).The claim in the following proposition is a trivial fact about preservation of thelimit.Proposition 1 (see [14]). Let I1, I2 be admissible ideals such that I1 ⊆ I2. IfI1 − limxn = L, then I2 − limxn = L.Whenever 0 < q < q′ < 1, we getIfin  I(q)c  I(q′)c ⊆ Ic ⊆ Id. (1)For a function h : R+ → R+, such that∑n∈N h(n) = ∞ and∑n∈∅ h(n) = 0,an ideal Ih ={A ⊂ N :∑n∈A h(n) < ∞}is called a summable ideal. For anyfunction h, the ideal Ih is admissible, so Ifin ⊆ Ih.Another type of convergence related to an ideal I, the so-called I*-convergence,was defined in papers [13] and [14].Definition 2. Let I be an admissible ideal on N. A sequence (xn) of real (complex)numbers is said to be I∗-convergent to L if there exists a set H ∈ I such that forM = N \H = {m1 < m2 < · · · } we have limk→∞ xmk = L, where the limit is in theusual sense.It is easy to see that for an admissible ideal I we have that I∗-convergenceimplies I-convergence. The converse is not true (see [14], where the authors givea characterization of ideals I, for which I- and I∗-convergence are equivalent bymeans of the property (AP)).Definition 3. An ideal (not necessarily admissible) I ⊂ 2N is said to satisfy thecondition (AP) if for every countable family of mutually disjoint sets {A1, A2, . . . }belonging to I there exists a countable family of sets {B1, B2, . . . } such that thesymmetric difference Aj△Bj is finite for j ∈ N and B =∪∞j=1 Bj ∈ I.Ih-convergence and convergence of positive series 3The property (AP) is similar to the property (APO) (see [6, 9] and [18]). Allideals in (1) have the property (AP). There exist many examples of an ideal thatdoes not have the property (AP) (see e.g. [3, 14]).Proposition 2 (see [14]). The statement I∗−limxn = L follows from I−limxn = Lif and only if I satisfies the property (AP).An ideal I (not necessarily admissible) is called a P-ideal if for each sequence(An) of sets belonging to I there exists a set A∞ ∈ I such that An \A∞ is finite forall n ∈ N.The notions of P-ideal and ideal with the (AP) property coincide (see [4]).In [17], Olivier proved the so-called Olivier’s Theorem about the speed of con-vergence to zero of the terms of convergent positive series with nonincreasing terms.Specifically, if (an) is a nonincreasing positive sequence and∑∞n=1 an < ∞, thenlimn→∞ nan = 0 (see also [1, 12]). In [20], authors made a remark that themonotonicity condition in Olivier’s theorem can be dropped if the convergence ofthe sequence (nan) is weakened by means of the notion of I-convergence. Theyproved that for every positive real sequence (an) such that∑∞n=1 an < ∞, we haveIc − limnan = 0.In [11], there is a similar result for the ideals I(q)c (0 < q ≤ 1). For every positivereal sequence (an) such that∑∞n=1 aqn < ∞ for 0 < q ≤ 1, we have I(q)c −limnan = 0.The stronger condition of convergence of positive series also results in the strongerconvergence property of the summands.Results of this type are called a modified Olivier’s theorem. In [2, 7, 15] and[16], there is an extension of the results in [20]. Moreover, in [16], there is a nicehistorical context of the object of our research.In connection with the above results, we will study the properties of summableideals Ih for a function h : R+ → R+ such that∑n∈N h(n) = ∞. We will showthat the notions Ih- and Ih∗-convergence are equivalent. It is clear that a modifiedOlivier’s theorem is in general not valid for summable ideals.If we limit ourselves to a large class of ideals Igc for a function g : R+ → R+ suchthat∑n∈N1g(n) = ∞ and Igc ={A ⊂ N :∑a∈A1g(a) < ∞}we will find sufficientconditions for the real function g for the modified Olivier’s Theorem to remain valid.2. Olivier’s theorem for ideals IgcFirst of all, we prove some properties of summable ideals. Let h : R+ → R+ be afunction with the following properties:∑n∈Nh(n) = ∞ and∑n∈∅h(n) = 0.Then the systemIh ={A ⊂ N :∑a∈Ah(a) < ∞}4 V. Baláž, A. Maťašovský and T. Visnyaiis an admissible ideal, so Ifin ⊆ Ih. The ideal Ih is called a summable ideal. It iseasy to see that for a constant function h(x) = c, x ∈ R+ we have Ifin = Ih, andwe also obtain the same for function h(x) = x, x ∈ R+.More interesting for our purposes are the admissible ideals Ih such that Ih ̸=Ifin, i.e., they contain an infinite subset of N.The following theorem gives a characterization of such ideals.Theorem 1. Ih ̸= Ifin if and only if lim infn→∞ h(n) = 0.Proof. Suppose Ih ̸= Ifin. Then there exists an infinite set M = {m1 < m2 <· · · }  N such that∞∑k=1h(mk) < ∞.From this we see that limk→∞ h(mk) = 0; since h is positive, we havelim infn→∞ h(n) = 0.Suppose that lim infn→∞ h(n) = 0. Then there exists a set M = {m1 < m2 <· · · } ⊂ N such that limk→∞ h(mk) = 0. It means that we can construct an infiniteset M ′ = {mk1 < mk2 < · · · } ⊆ M with property h(mki) < 12i for every i ∈ N.Consequently, we have ∑mki∈M ′h(mki) <∞∑i=112i,therefore, the infinite set M ′ belongs to Ih and so Ih ̸= Ifin.There exist positive functions g, h : R+ → R+ such that g ̸= h and Ig = Ih. Thefollowing theorem gives sufficient conditions for functions g, h : R+ → R+ to validIg = Ih.Theorem 2. Let g, h : R+ → R+ such that∑∞n=1 g(n) =∑∞n=1 h(n) = ∞. If0 < lim infn→∞g(n)h(n)≤ lim supn→∞g(n)h(n)< ∞,then Ig = Ih.Proof. The condition0 < lim supn→∞g(n)h(n)< ∞implies that there exists such real number K > 0 that for every n ∈ N we have0 <g(n)h(n)≤ K,therefore, g(n) ≤ Kh(n). Let M ∈ Ih. Then∑n∈M h(n) < ∞. Immediately, wehave ∑n∈Mg(n) ≤ K∑n∈Mh(n) < ∞,Ih-convergence and convergence of positive series 5therefore, M ∈ Ig and so Ih ⊆ Ig.Analogously using the condition0 < lim infn→∞g(n)h(n)< ∞,we obtain Ig ⊆ Ih.Problem 1. It would also be interesting to have the necessary conditions for thefunctions g, h : R+ → R+ such that Ig = Ih.The next theorem shows that Ih- and Ih∗-convergence are equivalent. See also[10], where it is proved that each summable ideal is P-ideal, thus it has the property(AP) that is a sufficient and necessary condition for Ih- and Ih∗-convergence to beequivalent.Theorem 3. Let h : R+ → R+ be a real function. Then Ih- and Ih∗-convergencecoincide.Proof. It sufficies to show that for any sequence (xn) of real numbers such thatIh− limxn = L, there exists a set M = {m1 < m2 < · · · } ⊆ N such that N\M ∈ Ihand limk→∞ xmk = L. Without loss of generality, we can assume that (xn) is notconvergent in the usual sense, but it is Ih-convergent. For any positive integer k,let εk =12kandAk ={n ∈ N : |xn − L| ≥12k}.It is clear that A1 ⊆ A2 ⊆ · · · ⊆ An ⊆ · · · , and there exists n0 ∈ N such that An0 isan infinite set. As Ih − limxn = L, we have Ak ∈ Ih, i.e.,∑n∈Ak h(n) < ∞.Therefore, there exists an infinite sequence n1 < n2 < · · · < nk < · · · of positiveintegers such that for every k = 1, 2, . . . we have∑n>nkn∈Akh(n) <12k.PutH =∞∪n=1[(nk, nk+1⟩ ∩Ak] .Then ∑n∈Hh(n) ≤∑n>n1n∈A1h(n) +∑n>n2n∈A2h(n) + · · ·+∑n>nkn∈Akh(n) + · · ·<12+122+ · · ·+ 12k+ · · ·< ∞.Thus, H ∈ Ih. Put M = N \H = {m1 < m2 < · · · < mk < · · · } and we show thatlimk→∞ xmk = L.6 V. Baláž, A. Maťašovský and T. VisnyaiLet ε > 0. Choose k0 ∈ N such that 12k0 < ε. Let mk > mk0 . Then mk belongsto some interval (nj , nj+1⟩ where j ≥ k0, and does not belong to N \ Aj (j ≥ k0).Hence mk belongs to N \ Aj and then |xmk − L| < ε for every mk > mk0 , thuslimk→∞ xmk = L.Corollary 1. Ideals Ih for a real function h : R+ → R+ have the property (AP).The next proposition shows that all bounded real sequences are not Ih-conver-gent.Proposition 3. Let h : R+ → R+. Then there exists a bounded real sequence (xn)that is not Ih-convergent.Proof. Since∑n∈N h(n) = ∞, there exists a decomposition of N into two sets N1and N2 such that ∑n∈N1h(n) =∑n∈N2h(n) = ∞.For instance, let (ni) be a sequence of nonnegative integers such thath(ni−1 + 1) + h(ni−1 + 2) + · · ·+ h(ni) > 1.DefineN1 =∪i∈N0i is odd{n : ni−1 < n ≤ ni},N2 =∪i∈N0i is even{n : ni−1 < n ≤ ni}.It is clear that N1, N2 /∈ Ih.Define a sequence (xn) as follows:xn ={0 if n ∈ N1,1 if n ∈ N2.The sequence (xn) is real bounded sequence which is not Ih-convergent.Corollary 2. An ideal Ih for any real function h : R+ → R+ is not a maximalideal.Proof. It follows from Theorem 2.2 in [13] that an admissible ideal I is the maximalideal if and only if each bounded real sequence (xn) is I-convergent. On the basisof the previous proposition, we have a contradiction.It is a natural question whether summable ideals Ih for a function h : R+ → R+can be used in a modified Olivier’s theorem in the following way:If∑n∈N h(an) is a convergent positive series for a function h : R+ → R+ and fora positive sequence (an), then Ih − limnan = 0.Ih-convergence and convergence of positive series 7It is easy to see that such modified Olivier’s theorem is not fulfilled in general.Consider a function h : R+ → R+, h(x) = x2 and the sequence (an), an = 1n forn ∈ N.Let g : R+ → R+ be a function such that∑n∈N1g(n)= ∞ and∑n∈∅1g(n)= 0. (2)Then the system of subsets of N, which denotes Igc ={A ⊂ N :∑n∈A1g(n) < ∞}is again an admissible ideal. Ideals Igc seem to be more convenient for a modifiedOlivier’s theorem.If we put a function g : R+ → R+, g(x) = x, we have the same result as in [20]for a function g(x) = xq for 0 < q ≤ 1 we obtain the same result as in [11].The following example shows that a modified Olivier’s theorem is not valid ingeneral for an arbitrary function g : R+ → R+ and an associated ideal Igc with thefunction g having properties (2).Example 1. Put g : R+ → R+, g(x) = log2(x+1). It is easy to see that the functiong is an increasing function such that∞∑n=11log2(n+ 1)= ∞ and lim infn→∞1log2(n+ 1)= 0.We show only that∑∞n=11log2(n+1)= ∞. It is easy to see that for all x > 1 wehave log2(x + 1) < x, and so1x <1log2(x+1). Using integrals for the last inequality,we obtain∞ =∫ ∞11xdx <∫ ∞11log2(x+ 1)dx.Hence∑∞n=11log2(n+1)= ∞. The idealI log2(x+1)c ={A ⊂ N :∑a∈A1log2(a+ 1)< ∞}is the admissible ideal, for which a modified Olivier’s theorem is not valid. It sufficiesto find a positive sequence (an) such that∑∞n=1 log2(an + 1) < ∞, but Ilog2(x+1)c −limnan ̸= 0. Take the set B ={2k − 1 : k ∈ N}and consider the following positivesequence (an):an ={1n if n ∈ B,12n if n ∈ N \B.Let us count∞∑n=1log2(n+ 1) =∑n∈Blog2(n+ 1) +∑n∈N\Blog2(n+ 1)=∞∑k=1log2(12k − 1+ 1)+∑n∈N\Blog2(12n+ 1).8 V. Baláž, A. Maťašovský and T. VisnyaiFirst, we show that the series∑∞k=1 log2(12k−1 + 1)is convergent. From the in-equality0 < log2(x+ 1) < 2x,for all x ∈ R+ we havelog2(12k − 1+ 1)<22k − 1.Since the series∑∞k=112k−1 is convergent, we also see that the series∑∞k=1 log2(12k−1 + 1)is convergent.In the same way, we also show convergence of the series∑n∈N\B log2(12n + 1).We will show that I log2(x+1)c − limnan ̸= 0. By using Definition 1 for any ideal I,we have that a real sequence (xn) is I-convergent to zero if for each ε > 0 the setAε = {n ∈ N : |xn| ≥ ε} belongs to the ideal I. In our case, it means that for ε = 1and the sequence (nan) the set Aε=1 = {n ∈ N : nan ≥ 1} belongs to I log2(x+1)c . Itsufficies to realize that Aε=1 ⊇ B and B /∈ I log2(x+1)c . Count∑n∈B1log2(x+ 1)=∞∑k=11log2(2k − 1 + 1)=∞∑k=11log2 2k=∞∑k=11k= ∞.The next theorem gives a sufficient condition for a real function g : R+ → R+such that a modified Olivier’s theorem is true for an associated ideal Igc with thefunction g.Theorem 4. Let a function g : R+ → R+ have the following properties:(i) g is nondecreasing,(ii) g(nt) ≤ g(n)g(t) for all n ∈ N and t ∈ R+.If∑∞n=1 g(an) is a convergent series for a positive sequence (an), then Igc −limnan =0.Proof. We proceed by contradiction. Then there exists a positive sequence (an)with∑∞n=1 g(an) < ∞ such that the equality Igc − limnan = 0 does not hold. Thenthere exists ε0 > 0 for which Aε0 = {n ∈ N : nan ≥ ε0} /∈ Igc . Hence from thedefinition of ideal Igc we get∑n∈Aε01g(n) = ∞. For n ∈ Aε0 we have nan ≥ ε0.Using properties (i) and (ii) we have0 < g(ε0) ≤ g(nan) ≤ g(n)g(an),g(ε0)1g(n)≤ g(an) for every n ∈ Aε0 .Therefore,∞ = g(ε0)∑n∈Aε01g(n)≤∑n∈Aε0g(an).Therefore, it must also be∑∞n=1 g(an) = ∞, and this is a contradiction.Ih-convergence and convergence of positive series 9Problem 2. To find the necessary condition for a function g : R+ → R+ such thata modified Olivier’s theorem is true for an associated ideal Igc with the function g.References[1] N. H. Abel, L. Olivier, Note sur le mémoire de mr. l. olivier no. 4. du second tomede ce journal, ayant pour titre “remarques sur les séries infinies et leur convergence”,J. Reine Angew. Math. 3(1828), 79–82.[2] V. Baláž, K. Liptai, J. T. Tóth, T. Visnyai, Convergence of positive series andideal convergence, Ann. Math. Inform. 52(2020), 19–30.[3] V. Baláž, T. Visnyai, I-convergence of Arithmetical Functions, Number Theory andIts Applications, IntechOpen, London, 2020.[4] M. Balcerzak, K. Dems, A. 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(I^h)-convergence and convergence of positive series

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