A Classical Olivier’s Theorem and Statistical Convergence

L. Olivier proved in 1827 the classical result about the speed of convergence to zero of the terms of a convergent series with positive and decreasing terms. We prove that this result remains true if we omit the monotonicity of the terms of the series when the limit operation is replaced by the statistical limit, or some generalizations of this concept. Résumé. L. Olivier démontrait en 1827 un résultat classique sur la vitesse de convergence vers zéro d’une série convergente à termes positifs décroissants. Nous démontrons que ce résultat reste valable si nous omettons la monotonie des termes de la série, en remplaçant l’opération limite par la limite statistique ou encore par des généralisations de ce concept.


Introduction
The above mentioned result of L. Olivier was published in [7] p. 39 (see also [5] p. 125) and claims that if a n ≥ a n+1 > 0, (n = 1, 2, . . . ) and Remark: As the unknown referee pointed out the example can be strengthened taking a n = log n n if n = k 2 , in which case the sequence (na 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 paper [2], (see also [3], [10], and [9]) Definition: We say that a sequence (x n ) n≥1 (of real or complex 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 asymptotic density, i.e. the limit 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 ∞ n=1 a n < +∞, (a n > 0, n = 1, 2, . . . ) without the monotonicity assumption on the sequence (a n ) n≥1 .
The notion of statistical convergence was generalized using the concept of an admissible ideal of subsets of positive integers N = {1, 2, . . .}, that is ⊆ P(N) and is additive (i.e.A, B ∈ ⇒ A ∪ B ∈ ), hereditary (i.e.B ⊂ A ∈ ⇒ B ∈ ), containing all singletons and not containing N.
Definition: (See [6].)We say that a sequence (x n ) n≥1 -converges to a number L and we write -lim x n = L, if for each ε > 0 the set A(ε) := {n : |x n − L| ≥ ε} belongs to the ideal .An admissible ideal is for example: Let us note that f -lim x n = L means the same as lim n→∞ x n = L.Some admissible ideals can be obtained using various concepts of density of sets A ⊆ N. Using the asymptotic density defined above we obtain the ideal Obviously d -convergence means the statistical convergence.Another type of density is the logarithmic density defined by means of lower and upper logarithmic density of a set A ⊆ N: Using the logarithmic density we can define the ideal A little bit more complicated is the concept of the uniform density.For any s is similar.Let us choose a fixed p ∈ N. Then for any s ∈ N there exists t s ≥ 1 such that β s = A(t s + 1, t s + s) and simultaneously β p ≤ A(t + 1, t + p) for every t ≥ t s .For any s ∈ N there exist unique integer numbers k s , r s ≥ 0 such that s = k s p + r s with 0 ≤ r s ≤ p − 1.Then we have (with the convention A(x, y) = 0 if y < x) Hence βs s ≥ ksβp ksp+rs and when s → ∞ then also k s → ∞ and so for fixed p we have lim ks→∞ The following relations between these densities can be verified (cf.[1], [4]): Consequently we get the chain of inclusions for the above defined ideals: The ideal which will play an important role in the main theorem is the following one It is well known (see [8]) that

Main Results
The above mentioned Olivier's result can be formulated in the terms of fconvergence as follows.If In the sequel we are going to study the ideals with the following property: From Example 1 we can conclude that the ideal f does not have the property 2.1.Let us denote by S(T ) the class of all admissible ideals , with the property 2.1.So we have that f / ∈ S(T ).The following theorem claims a more useful fact.

Theorem 2.1: Ideal c is an element of S(T ).
Proof: We proceed by contradiction.Let Then there exist numbers a n < +∞ such that the equality 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 } / ∈ c .Hence from the definition of the ideal c we get n∈A(ε0) n −1 = +∞.For n ∈ A(ε 0 ) we have na n ≥ ε 0 and so a n ≥ ε 0 n for every n ∈ A(ε 0 ).
Using this and the comparison criterion for infinite series we get n∈A(ε0) So it must be also a n = +∞ and this is a contradiction.
The claim in the following lemma is a trivial fact about preservation of the limit.Lemma 2.2: Let 1 , 2 be admissible ideals such that 1 ⊆ 2 .If 1lim x n = L, then also 2 -limx n = L.
An obvious consequence of Lemma 2.2 is the following theorem.In connection with Theorem 2.6 it is important to know how many ideals have the property 2.1 and how many don't have this property.We know yet that the ideal f which yields usual convergence is not an element of S(T ).We give some more such examples.

. ).
Obviously A ∈ c .To prove that A / ∈ u let us determine the upper 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, . . .The next two propositions give an idea of how many admissible ideals are appropriate for the analog of Olivier's theorem and how many can not be used to obtain this analog.
can conclude that there exists the limit lim s→∞ βs s = sup p≥1 βp p .If u(A) = u(A) =: u(A) then the common value u(A) is called the uniform density of A (cf.[1]).Using the uniform density we can define the admissible ideal u := {A ⊆ N : u(A) = 0}.To compare the above defined ideals let us remember the lower and upper asymptotic density defined by d(A) := lim inf n→∞

Proof: 1 )
If ∈ S(T ) then after Theorem 2.5 we have ⊇ c .2) Let be an admissible ideal and c ⊆ .Due to Theorem 2.1 we have c ∈ S(T ) and the Corollary 2.4 yields ∈ S(T ).Remark: Referring to inclusions 1.1 and 1.2 we can claim that S(T ) contains as elements the ideals d and δ .Since the d -convergence is in fact the statistical convergence we get ∞ n=1 a n < +∞, (a n > 0) =⇒ lim-stat na n = 0.