International Journal of Analysis and Applications

Volume 18, Number 3 (2020), 332-336

URL: https://doi.org/10.28924/2291-8639

DOI: 10.28924/2291-8639-18-2020-332

A NOTE ON OLIVIER’S THEOREM AND CONVERGENCE IN ERDŐS-ULAM

DENSITY

JÓZSEF BUKOR∗

Department of Mathematics and Informatics, J. Selye University, 945 01 Komárno, Slovakia

∗Corresponding author: bukorj@ujs.sk

Abstract. Olivier’s Theorem says that if
∑

an is a convergent positive series and (an) is monotone de-

creasing, then nan → 0. Šalát and Toma [4] proved that the monotonicity condition can be omitted if

the convergence of (nan)n is replaced by the statistical convergence. The aim of this note is to give an

alternative proof and generalization of this result.

1. Introduction

A classical Olivier’s Theorem says that if
∑
an is a convergent positive series and (an) is monotone

decreasing, then nan → 0.

T. Šalát and V. Toma proved in 2003 [4] that the monotonicity condition in the above result can be

omitted if the convergence of (nan)n is replaced by the statistical convergence. This result was generalized

and extended by several authors, see e.g., [3] and [2].

The aim of this note is to give an alternative proof and a generalization of the result of Šalát and Toma,

and extend a result of Niculescu and Prǎjiturǎ (see [3], Theorem 6) which we recall later.

From now on, we call a positive function f : N → (0,∞) weight function (or Erdős-Ulam function) if it

satisfies
∞∑

n=1

f(n) = ∞ and lim
n→∞

f(n)∑n
j=1 f(j)

= 0 .

Received February 20th, 2020; accepted March 19th, 2020; published May 1st, 2020.

2010 Mathematics Subject Classification. 40A30, 40A35.

Key words and phrases. positive series; weighted density; convergence in density.

©2020 Authors retain the copyrights

of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

332

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-18-2020-332


Int. J. Anal. Appl. 18 (3) (2020) 333

With respect to a weight function f the f-weighted densities are defined as follows. For A ⊂ N let

F(A,n) =

n∑
j=1

f(j) ·χA(j)∑n
j=1 f(j)

,

where χA denotes the characteristic function of A. Now we define the lower and upper f-densities of A by

df (A) = lim inf
n→∞

F(A,n) and df (A) = lim sup
n→∞

F(A,n),

respectively. In the case when df (A) = df (A) we say that A has the f-density property denoted by df (A).

Note that the asymptotic density corresponds to f(n) = 1, while the logarithmic density does to f(n) =

1/n. The logarithmic density is related to the asymptotic density via the inequalities

0 ≤ d1(A) ≤ d 1
n

(A) ≤ d 1
n

(A) ≤ d1(A) ≤ 1 .

Define the function f∗ by

f∗(n) =
f(n)∑n
j=1 f(j)

. (1.1)

The logarithmic density can be considered as a density derived from the asymptotic density by (1.1). This

method can be extended for an arbitrary weighted density given by the weight function f to provide a new

weight function f∗ (and, consequently, a new weighted density). Moreover, for arbitrary A ⊂ N we have

df (A) ≤ df∗(A) ≤ df∗(A) ≤ df (A) , (1.2)

see [1].

The concept of convergence in density is an extension of the concept of statistical convergence. A sequence

(an) converges to a number α in density df , which we denote as (df )– limn→∞an = α, provided the set

Aε = {n ∈ N : |an −α| ≥ ε}

has zero f-density, i.e., df (Aε) = 0.

Now, we can rewrite the result of Šalát and Toma as

if
∑

an is a convergent positive series, then (d1)– lim
n→∞

nan = 0. (1.3)

Niculescu and Prǎjiturǎ [3] studied an analogous question for the harmonic density. They stated that

if
∑

an is a convergent positive series, then (d 1
n

)– lim
n→∞

(n ln n)an = 0. (1.4)

We generalize these results above.



Int. J. Anal. Appl. 18 (3) (2020) 334

2. Results

In the proof of our theorem we will use the following observation.

Lemma 2.1. Let f be an Erdős-Ulam function and f∗ is defined by (1.1). Let A be an infinite set of positive

integers such that
∑

k∈A f
∗(k) is convergent. Then df (A) = 0.

Proof. From the assertion of the lemma df∗(A) = 0 follows immediately. But inequality (1.2) does not

give any information on the behavior of df (A). Taking into account that the upper density of a set does

not change by removing finitely many elements. This observation, together with the fact that the tail of a

convergent series tends to zero shows

df (A) = lim
n→∞

(
lim sup
m→∞

∑
k∈A∩[n,m] f(k)∑m

k=1 f(k)

)
≤ lim

n→∞

(
lim

m→∞

∑
k∈A∩[n,m]

f(k)∑k
j=1 f(j)

)

= lim
n→∞

(
lim

m→∞

∑
k∈A∩[n,m]

f∗(k)

)
≤ lim

n→∞

∑
k∈A∩[n,∞)

f∗(k) = 0.

�

Hence df (A) = 0.

Theorem 2.1. Let f be an Erdős-Ulam function. If
∑
an is a convergent positive series, then

(df )– lim
n→∞

∑n
k=1 f(k)

f(n)
an = 0 . (2.1)

Proof. Fix ε > 0, and consider the set

Aε = {n ∈ N :
∑n

k=1 f(k)

f(n)
an ≥ ε} .

Since

ε
∑
n∈Aε

f∗(n) = ε

∑
n∈Aε

f(n)∑n
k=1 f(k)

≤
∑
n∈Aε

an ≤
∑
n∈N

an < ∞,

applying Lemma 2.1 we immediately get that the set Aε has zero f-density. Then (2.1) holds and the proof

is completed. �

Corollary 2.1. If we consider the asymptotic density in (2.1), then we conclude (1.3). Similarly, the

logarithmic density (if f(n) = 1/n) leads to (1.4). For f(n) = 1/(n ln n) (the case of loglog-density), we

obtain

if
∑

an is a convergent positive series, then (d 1
n ln n

)– lim
n→∞

n(ln n)(ln ln n)an = 0.

Roughly speaking, if
∑
an is a convergent positive series, then the fast growing of the weight function f

guarantees a less speed convergence of (an) to zero in density df .



Int. J. Anal. Appl. 18 (3) (2020) 335

For example, let f(n) = e
√
n/(2
√
n). In this case

∑n
k=1 f(k) ∼ e

√
n and we have

if
∑

an is a convergent positive series, then (df )– lim
n→∞

√
nan = 0.

Next, we show that (1.3) is best possible in the sense that we cannot replace (d1)– limn→∞nan = 0 with

(d1)– limn→∞nωnan = 0, where ωn is an arbitrary sequence tending to infinity.

Theorem 2.2. Let (ωn) be an increasing sequence, tending to infinity. Then there exists a sequence (an) of

positive terms, such that
∑
an converges and (d1)– limn→∞nωnan 6= 0.

Proof. The construction of (an) is based on the fact that

lim
m→∞

2m∑
k=m

1

kωk
≤ lim

m→∞

1

ωm

2m∑
k=m

1

k
= lim

m→∞

ln 2

ωm
= 0 . (2.2)

Using (2.2) we are able to define an increasing sequence (mi) for that

mi+1 > 2mi and

2mi∑
k=mi

1

kωk
<

1

2i
, i = 1, 2, . . . .

Define the sequence (an) as

an =




1
n2ωn

if n ∈ N r
∞⋃
i=1

[mi, 2mi]

1
nωn

if n ∈
∞⋃
i=1

[mi, 2mi].

Then
∑
an converges since

∞∑
n=1

an =
∑

n∈Nr∪∞
i=1

[mi,2mi]

1

n2ωn
+

∑
n∈∪∞

i=1
[mi,2mi]

1

nωn

≤
∞∑

n=1

1

n2
+

∞∑
i=1

k=2mi∑
k=mi

1

kωk
<
π2

6
+

∞∑
i=1

1

2i
=
π2

6
+ 1 .

We are going to show that (d1)– limn→∞nωnan = 0 fails. Fix ε ∈ (0, 1) and consider the set

Aε = {n ∈ N : nωnan ≥ ε} .

Then for any n ∈ [mi, 2mi] we have nωnan = 1 and therefore the set Aε does not have zero asymptotic

density. �

Conflicts of Interest: The author(s) declare that there are no conflicts of interest regarding the publication

of this paper.



Int. J. Anal. Appl. 18 (3) (2020) 336

References

[1] J. Bukor, F. Filip and J. T. Tóth, Sets with countably infinitely many prescribed weighted densities, Rocky Mountain J.

Math.(to appear).

[2] A. Faisant,G. Grekos and L. Mǐśık, Some generalizations of Olivier’s theorem, Math. Bohem. 141 (4) (2016), 483–494.

[3] C. P. Niculescu and G. T. Prǎjiturǎ, Some open problems concerning the convergence of positive series, Ann. Acad. Rom.

Sci. Ser. Math. Appl. 6 (1) (2014), 92–107.

[4] T. Šalát and V. Toma, A Classical Olivier’s Theorem and Statistical Convergence, Ann. Math. Blaise Pascal, 10 (2) (2003),

305–313.


	1. Introduction
	2. Results
	References