On Faster Convergent Infinite Series

Abstract

Suffcient conditions, necessary conditions for faster convergent infinite series, faster 𝜏-convergent infinite series are studied. The faster convergence of infinite series of Kummer's type is proved.

1. Introduction

There have been papers devoted to the study of faster convergence of sequences. Certain methods concerning acceleration of convergence of sequences of partial sums of fixed series using linear or nonlinear transformations of partial sums of series are studied in [1]. The acceleration field of subsequence matrix transformations with respect to the convergence rate of the sequence being accelerated are studied in [2]. In [3], it is discussed a class of methods for summing sequences which are generalizations of a method due to Salzer [4], which accelerate some convergent sequences especially monotone sequences. In [5], is characterized the summability field of a matrix 𝐴 by showing 𝐴 is convergence preserving over the set of all sequences which converge faster than some fixed sequence 𝑥, 𝐴 is convergence preserving over the set of all sequences, or 𝐴 only preserves the limit of a set of constant sequences. Statistical acceleration convergence of sequences was discussed in [6]. The notion of faster convergent series with positive terms is defined in [7] and the notion of 𝜏-convergent series is defined in [8–10]. The statistical convergence of infinite series as a special case of 𝜏-convergence of infinite series is discussed in [11–14].

In this paper, some questions related to sufficient conditions and necessary conditions for faster convergent infinite series are studied, faster 𝜏-convergent series are defined and studied, and faster convergence of series of Kummer's type is proved. In [7, page 146], it is mentioned the Kummer's result: if ∑^∞𝑛=1𝑎_𝑛 is convergent series with positive terms and with an unknown sum 𝑎, ∑^∞𝑛=1𝑐_𝑛 is a convergent series with positive terms and with a known sum 𝑐, and lim_𝑛→∞𝑎_𝑛/𝑐_𝑛=𝑝>0, then the Kummer series ∑^∞𝑛=1𝑏_𝑛 has the same sum as ∑^∞𝑛=1𝑎_𝑛 and is a faster convergent series than ∑^∞𝑛=1𝑎_𝑛, if all terms of ∑^∞𝑛=1𝑏_𝑛 are positive. Hence we can calculate the unknown sum 𝑎 faster by summation terms of the Kummer series ∑^∞𝑛=1𝑏_𝑛, which is constructed by ∑^∞𝑛=1𝑎_𝑛, ∑^∞𝑛=1𝑐_𝑛, and 𝑝. In Lemma 4.1, we proved a faster convergence of the Kummer series ∑^∞𝑛=1𝑏_𝑛 to the unknown sum 𝑎 of ∑^∞𝑛=1𝑎_𝑛 without conditions of positivity of 𝑝 and terms of ∑^∞𝑛=1𝑎_𝑛,∑^∞𝑛=1𝑏_𝑛,∑^∞𝑛=1𝑐_𝑛.

We denote by ℕ the set of all positive integers and by ℝ the set of all real numbers.

Definition 1.1 (See [2]). Let ∑^∞𝑛=1𝑎_𝑛, ∑^∞𝑛=1𝑏_𝑛 be convergent real series with the same sum, with nonzero terms and such that 𝑏_𝑛+𝑏_𝑛+1+…≠0,𝑛∈ℕ. The series ∑^∞𝑛=1𝑎_𝑛 is called faster convergent than ∑^∞𝑛=1𝑏_𝑛 if lim_𝑛→∞(𝑎_𝑛+𝑎_𝑛+1+…)/(𝑏_𝑛+𝑏_𝑛+1+…)=0.

Lemma 1.2 (See [7]). Let ∑^∞𝑛=1𝑎_𝑛, ∑^∞𝑛=1𝑏_𝑛 be convergent real series with positive terms and with the same sum. If lim_𝑛→∞(𝑎_𝑛/𝑏_𝑛)=0, then lim_𝑛→∞(𝑎_𝑛+𝑎_𝑛+1+…)/(𝑏_𝑛+𝑏_𝑛+1+…)=0.

In what follows, we do not assume equality of sum of convergent series ∑^∞𝑛=1𝑎_𝑛,∑^∞𝑛=1𝑏_𝑛 because we can construct series ∑^∞𝑛=1𝑎^∗𝑛,∑^∞𝑛=1𝑏^∗𝑛, where 𝑎^∗𝑛=𝑎_𝑛 and 𝑏^∗𝑛=𝑏_𝑛 for 𝑛≥2 with the same sum.

2. Faster Convergent Series

Lemma 2.1. Let ∑^∞𝑛=1𝑎_𝑛, ∑^∞𝑛=1𝑏_𝑛 be convergent real series with positive terms. If lim_𝑛→∞𝑎_𝑛+𝑎_𝑛+1+…𝑏_𝑛+𝑏_𝑛+1+…=0,thenliminf^𝑛→∞𝑎_𝑛𝑏_𝑛=0.(2.1)

Proof. By the way of contradiction, we suppose that liminf_𝑛→∞(𝑎_𝑛/𝑏_𝑛)=𝑞>0, then there exists 𝑛₀∈ℕ such that for every 𝑛>𝑛₀ we have 0<𝑞^⋆<𝑎_𝑛/𝑏_𝑛, where 0<𝑞^⋆<𝑞. From this follows 𝑞^⋆(𝑏_𝑛+𝑏_𝑛+1+…)<(𝑎_𝑛+𝑎_𝑛+1+…) for 𝑛>𝑛₀, which is a contradiction with lim_𝑛→∞(𝑎_𝑛+𝑎_𝑛+1+…)/(𝑏_𝑛+𝑏_𝑛+1+…)=0, similarly for 𝑞=∞.

Lemma 2.2. Let ∑^∞𝑛=1𝑎_𝑛, ∑^∞𝑛=1𝑏_𝑛 be convergent real series with positive terms. Let lim_𝑛→∞(𝑎_𝑛/𝑏_𝑛) exist. Then lim_𝑛→∞𝑎_𝑛𝑏_𝑛=0ifflim_𝑛→∞𝑎_𝑛+𝑎_𝑛+1+…𝑏_𝑛+𝑏_𝑛+1+…=0.(2.2)

The next example shows that under the conditions of Lemma 2.1, for all 0<𝑟≤∞, there exist a series ∑^∞𝑛=1𝑎_𝑛,∑^∞𝑛=1𝑏_𝑛 such that limsup_𝑛→∞(𝑎_𝑛/𝑏_𝑛)=𝑟.

Example 2.3. Let {𝛼_𝑛;𝑛∈ℕ} be any sequence of real positive numbers which satisfy 𝛼_𝑛≤1/𝑛⁴ for 𝑛≥1. We define the series ∑^∞𝑛=1𝑎_𝑛=𝑟𝛼₁+𝛼₁/1²+𝑟𝛼₂+𝛼₂/2²+…+𝑟𝛼_𝑘+𝛼_𝑘/𝑘²+… and ∑^∞𝑛=1. Then we have𝑏_𝑛 If =𝛼₁+1²𝛼₁+𝛼₂, then +2²𝛼₂+…+

Remark 2.4. It is evident that in general from 𝛼_𝑘+𝑘²𝛼_𝑘+… does not follow lim_𝑛→∞𝑎_2𝑛−1+𝑎_2𝑛+…𝑏_2𝑛−1+𝑏_2𝑛+…=lim^𝑛→∞𝑟𝛼_𝑛+𝛼_𝑛+1+…+1/𝑛²𝛼^𝑛+𝛼_𝑛+1+…lim_𝑛→∞𝛼^𝑛+𝛼_𝑛+1+…+𝑛²𝛼^𝑛+𝛼_𝑛+1+…=0,(2.3)𝑎_2𝑛+𝑎_2𝑛+1+…𝑏_2𝑛+𝑏_2𝑛+1+…=lim^𝑛→∞𝑟𝛼_𝑛+1+𝛼_𝑛+2+…+1/𝑛²𝛼^𝑛+𝛼_𝑛+1+…𝛼^𝑛+1+𝛼_𝑛+2+…lim+_𝑛→∞𝑛²𝛼^𝑛+𝛼_𝑛+1+…=0,(1)𝑎_2𝑛−1𝑏_2𝑛−1=𝑟,lim_𝑛→∞𝑎_2𝑛𝑏_2𝑛=0.(2) For example, if we put ∑^∞𝑛=1𝑎_𝑛=𝛼₁+𝛼₁/1²+2𝛼₂+𝛼₂/2²+…+𝑘𝛼_𝑘+𝛼_𝑘/𝑘²+…lim_𝑛→∞(𝑎_2𝑛−1/𝑏_2𝑛−1)=∞. and lim_𝑛→∞(𝑎_𝑛+𝑎_𝑛+1+…)/(𝑏_𝑛+𝑏_𝑛+1+…)=0liminf_𝑛→∞(𝑎_𝑛/𝑏_𝑛)=0., we obtain 𝑎_2𝑛=𝑏_2𝑛=1/(𝑛(𝑛+1)), and 𝑛∈ℕ, is the set of all cluster points of the sequence 𝑎_2𝑛−1=−𝑏_2𝑛−1=−1/(𝑛(𝑛+1)),

In general, the condition 𝑛∈ℕ does not imply the condition lim_𝑛→∞(𝑎_𝑛+𝑎_𝑛+1+…)/(𝑏_𝑛+𝑏_𝑛+1+…)=0 as it follows from the next example.

Example 2.5. Let {−1,1} be a sequence such that {𝑎_𝑛/𝑏_𝑛;𝑛∈ℕ}.lim_𝑛→∞(𝑎_𝑛/𝑏_𝑛)=0
lim_𝑛→∞(𝑎_𝑛+𝑎_𝑛+1+…)/(𝑏_𝑛+𝑏_𝑛+1+…)=0. We define convergent series {𝛼_𝑛;𝑛∈ℕ} and 𝛼_𝑛=𝑞^𝑛,. From the definition, we obtain 𝑛∈ℕ, It is obvious that 0<𝑞<1/2 and ∑^∞𝑛=1𝑎_𝑛=𝛼₁+𝛼₁+𝛼₂/2²+𝛼₂/2²+…+𝛼_𝑘/𝑘²+𝛼_𝑘/𝑘²+… From ∑^∞𝑛=1𝑏_𝑛=𝛼₁−𝛼₁/2+𝛼₂/3−𝛼₂/4+…+𝛼_𝑘/(2𝑘−1)−𝛼_𝑘/2𝑘+…𝑏_2𝑛+𝑏_2𝑛+1+…=−𝛼^𝑛2𝑛+𝛼^𝑛+1<12𝑛+1(2𝑛+2+2𝑛+3𝛼^𝑛+22𝑛+4+…2𝑛−𝛼+𝛼^𝑛+1=1+𝛼_𝑛+2+…2𝑛−𝑞^𝑛+𝑞^𝑛+1+𝑞^𝑛+2+…=𝑞^𝑛2𝑞−12𝑛1−𝑞<0.(2.4) we have𝑏_2𝑛+1+𝑏_2𝑛+2+…>0Thus lim_𝑛→∞(𝑎_𝑛/𝑏_𝑛)=0.

Lemma 2.6. Let (2(𝑘+𝑚)−1)(2(𝑘+𝑚))>(𝑘+𝑚)²,, 𝑘,𝑚∈ℕ be convergent real series with nonzero terms. Let 𝑎_2𝑘−1+𝑎_2𝑘+…𝑏_2𝑘−1+𝑏_2𝑘+…=2𝛼^𝑘/𝑘²+𝛼_𝑘+1/(𝑘+1)²+…𝛼_𝑘/(2𝑘(2𝑘−1))+𝛼_𝑘+1/(2𝑘+1)(2𝑘+2)+…>2for𝑘≥1.(2.5) for all lim_𝑛→∞(𝑎_𝑛+𝑎_𝑛+1+…)/(𝑏_𝑛+𝑏_𝑛+1+…)≠0.. Let ∑^∞𝑛=1𝑎_𝑛∑^∞𝑛=1𝑏_𝑛𝑏_𝑛+𝑏_𝑛+1+…≠0𝑛∈ℕ then (1)𝑙_𝑖(𝑎)=liminf_𝑛→∞|1+𝑎_𝑛+1/𝑎_𝑛+𝑎_𝑛+2/𝑎_𝑛+…|,(2)𝑙_𝑠(𝑎)=limsup_𝑛→∞|1+𝑎_𝑛+1/𝑎_𝑛+𝑎_𝑛+2/𝑎_𝑛+…|,(3)𝑙_𝑖(𝑏)=liminf_𝑛→∞ if and only if |1+𝑏_𝑛+1/𝑏_𝑛+𝑏_𝑛+2/𝑏_𝑛+…|,

Proof. For every 𝑙_𝑠(𝑏)=limsup_𝑛→∞|1+𝑏_𝑛+1/𝑏_𝑛+𝑏_𝑛+2/𝑏_𝑛+…|,, we haveif𝑙_𝑠(𝑎)<∞,𝑙_𝑖(𝑏)>0,lim_𝑛→∞(𝑎_𝑛/𝑏_𝑛)=0,thenlim_𝑛→∞(𝑎_𝑛+𝑎_𝑛+1+…)/(𝑏_𝑛+𝑏_𝑛+1+From this follows our assertion.

Remark 2.7. The condition …)=0, can be satisfied, for example, if if𝑎_𝑛+𝑎_𝑛+1+…≠0forall𝑛∈ℕ,𝑙_𝑖(𝑎)>0,𝑙_𝑠(𝑏)<∞,lim_𝑛→∞(𝑎_𝑛+𝑎_𝑛+1+…)/(𝑏_𝑛+ In fact, if 𝑏_𝑛+1+…)=0,thenlim_𝑛→∞(𝑎_𝑛/𝑏_𝑛)=0, then there exists if𝑎_𝑛+𝑎_𝑛+1+…≠0forall𝑛∈ℕ,0<𝑙_𝑖(𝑎),𝑙_𝑠(𝑎)<∞,0<𝑙_𝑖(𝑏),𝑙_𝑠(𝑏)<∞, such that for every thenlim_𝑛→∞(𝑎_𝑛/𝑏_𝑛)=0, we havelim_𝑛→∞(𝑎_𝑛+𝑎_𝑛+1+…)/(𝑏_𝑛+𝑏_𝑛+1+…)=0.and thus 𝑛∈ℕ The condition |||𝑎_𝑛+𝑎_𝑛+1+…𝑏_𝑛+𝑏_𝑛+1+…|||=|||1+𝑎^𝑛+1/𝑎_𝑛+𝑎_𝑛+2/𝑎_𝑛+…||||||𝑏_𝑛=/|||𝑎1_𝑛++𝑎𝑏_𝑛+1𝑛+1/𝑎_𝑛+𝑏_𝑛+2/𝑎_𝑛+…|||/𝑎_𝑛+𝑎_𝑛+2/𝑎_𝑛+…||||||𝑏_𝑛/𝑎_𝑛||||||1+𝑏_𝑛+1=|||𝑎/𝑏_𝑛+𝑏_𝑛+2/𝑏_𝑛+…_𝑛|||/𝑏_𝑛||||||1+𝑎_𝑛+1/𝑎_𝑛+𝑎_𝑛+2/𝑎_𝑛+𝑎_𝑛+3/𝑎_𝑛+…||||||1+𝑏_𝑛+1/𝑏_𝑛+𝑏_𝑛+2/𝑏_𝑛+𝑏_𝑛+3/𝑏_𝑛+…|||.(2.6) can be satisfied, for example, if liminf_𝑛→∞|1+𝑏_𝑛+1/𝑏_𝑛+𝑏_𝑛+2/𝑏_𝑛+…|>0 Indeed, if limsup_𝑛→∞|𝑏_𝑛+1/𝑏_𝑛|<1/2. then there exists limsup_𝑛→∞|𝑏_𝑛+1/𝑏_𝑛|<𝑟<1/2. such that for every 𝑛₀∈ℕ, we have𝑛>𝑛₀Conversely, from the condition |||𝑏_𝑛+1𝑏_𝑛+𝑏^𝑛+2𝑏_𝑛+𝑏^𝑛+3𝑏_𝑛+…|||≤|||𝑏^𝑛+1𝑏_𝑛|||+|||𝑏_𝑛+2𝑏_𝑛+1||||||𝑏_𝑛+1𝑏_𝑛|||+|||𝑏_𝑛+3𝑏_𝑛+2||||||𝑏_𝑛+2𝑏_𝑛+1||||||𝑏_𝑛+1𝑏_𝑛≤𝑟+𝑟₂|||+…+𝑟³+…=𝑟1−𝑟,(2.7) need not follow the condition |1+𝑏_𝑛+1/𝑏_𝑛+𝑏_𝑛+2/𝑏_𝑛+…|≥(1−2𝑟)/(1−𝑟)>0. For example, if we put limsup_𝑛→∞|1+𝑎_𝑛+1/𝑎_𝑛+𝑎_𝑛+2/𝑎_𝑛+…|<∞, limsup_𝑛→∞|𝑎_𝑛+1/𝑎_𝑛|<1., limsup_𝑛→∞|𝑎_𝑛+1/𝑎_𝑛|=𝛼<1,, 𝑛₀∈ℕ then 𝑛>𝑛₀ is convergent series and |||1+𝑎_𝑛+1𝑎_𝑛+𝑎^𝑛+2𝑎_𝑛+…|||=|||1+𝑎^𝑛+1𝑎_𝑛+𝑎^𝑛+2𝑎_𝑛+1𝑎_𝑛+1𝑎_𝑛+𝑎^𝑛+3𝑎_𝑛+2𝑎_𝑛+2𝑎_𝑛+1𝑎_𝑛+1𝑎_𝑛≤1+𝛽+𝛽₂+…|||+𝛽³+…=11−𝛽<∞,where𝛼<𝛽<1.(2.8), liminf_𝑛→∞|1+𝑏_𝑛+1/𝑏_𝑛+𝑏_𝑛+2/𝑏_𝑛+…|>0

3. limsup_𝑛→∞|(𝑏_𝑛+1/𝑏_𝑛)|<1/2.-Convergent Series

Definition 3.1 (See [10]). We say that a sequence 𝑎₁≠0 has 𝑎_2𝑛=1/2𝑛-limit a real number 𝑎_2𝑛+1=−1/2𝑛 and we write 𝑛=1,2,…,-∑^∞𝑛=1𝑎_𝑛, if for each limsup_𝑛→∞|𝑎_𝑛+1/𝑎_𝑛|=1 the set limsup_𝑛→∞|1+𝑎_𝑛+1/𝑎_𝑛+𝑎_𝑛+2/𝑎_𝑛+…|=1. belongs to the ideal 𝜏, where {𝑎_𝑛;𝑛∈ℕ} is an admissible ideal of subsets of 𝜏 which is additive (if 𝐿, then 𝜏), hereditary (if lim_𝑛→∞𝑎_𝑛=𝐿, then 𝜀>0), containing all singletons and not containing 𝐴(𝜀)={𝑛;|𝑎_𝑛−𝐿|≥𝜀}.

We denote by 𝜏 the ideal of all finite subsets of 𝜏.

Definition 3.2 (See [10]). We say that ℕ𝐴,𝐵∈𝜏-converges to a real number 𝐴∪𝐵∈𝜏 and we write 𝐵⊂𝐴∈𝜏-𝐵∈𝜏 if for each ℕ the set 𝜏_𝑓 belongs to the ideal ℕ, where ∑^∞𝑛=1𝑎_𝑛 is an admissible ideal of subsets of 𝜏.

Definition 3.3. Let 𝐿, 𝜏 be ∑^∞𝑛=1𝑎_𝑛=𝐿-convergent real series with nonzero terms such that 𝜀>0𝐴(𝜀)={𝑛;|∑^𝑛𝑘=1𝑎_𝑘−𝐿|≥𝜀}. A series 𝜏 is called 𝜏-faster convergent than ℕ if ∑^∞𝑛=1𝑎_𝑛

Definition 3.4 (See [9]). Let ∑^∞𝑛=1𝑏_𝑛 be an admissible ideal of subsets of 𝜏. A number 𝑏_𝑛+𝑏_𝑛+1+…≠0, is said to be a 𝑛∈ℕ-cluster point of ∑^∞𝑛=1𝑎_𝑛 if for each 𝜏 the set ∑^∞𝑛=1𝑏_𝑛 is not from 𝜏-lim_𝑛→∞(𝑎_𝑛+1+𝑎_𝑛+2+…)/(𝑏_𝑛+1+𝑏_𝑛+2+…)=0.

Remark 3.5. Of course, if 𝜏 is an admissible ideal of subsets of ℕ, by 𝑥∈𝑅-cluster point of a real sequence 𝜏, we mean a number ∑^∞𝑛=1𝑥_𝑛, where for each 𝜀>0 the set {𝑛∈ℕ;|∑^𝑛𝑘=1𝑥_𝑘−𝑥|<𝜀} is not from 𝜏. Moreover, we say that 𝜏 (ℕ) is the 𝜏-cluster point of a real sequence {𝑥_𝑛;𝑛∈ℕ} if for each 𝑥∈ℝ (𝜀>0) the set {𝑛∈ℕ;|𝑥_𝑛−𝑥|<𝜀} (𝜏.) is not from ∞.

Remark 3.6. If −∞ is a real sequence, 𝜏 is an admissible ideal, and {𝑥_𝑛;𝑛∈ℕ}, then 𝑐>0, Indeed, if 𝑐<0 is bounded, then by [8] there exists a {𝑛∈ℕ;𝑥_𝑛>𝑐}-cluster point of {𝑛∈ℕ;𝑥_𝑛<𝑐} If 𝜏 is not bounded, then either {𝑥_𝑛;𝑛∈ℕ} or 𝜏 is 𝑋={𝑥;𝑥isa𝜏-clusterpointof{𝑥_𝑛;𝑛∈ℕ}}-cluster point of 𝑋≠∅. according to Remark 3.5 or there exists {𝑥_𝑛;𝑛∈ℕ} such that 𝜏 If for some {𝑥_𝑛;𝑛∈ℕ}.{𝑥_𝑛;𝑛∈ℕ}, then +∞ Consider the ideal −∞ Because 𝜏 is a bounded set, there exists {𝑥_𝑛;𝑛∈ℕ}𝑙∈ℝ such that {𝑛∈ℕ;|𝑥_𝑛|>𝑙}∈𝜏. is a 𝑙∈ℝ-cluster point of {𝑛∈ℕ;|𝑥_𝑛|>𝑙}∈𝜏 Let 𝐾_𝑙={𝑛∈ℕ;|𝑥_𝑛|≤𝑙}∉𝜏.. Since 𝜐={𝐾_𝑙∩𝐴;𝐴∈𝜏}. is a {𝑥_𝑛;𝑛∈𝐾_𝑙}-cluster point of 𝑥^⋆∈ℝ,, then the set |𝑥^⋆|≤𝑙 and then also is not from 𝑥^⋆. Since 𝜐 has a hereditary property, then {𝑥_𝑛;𝑛∈𝐾_𝑙}.. So 𝜀>0 is a cluster point of 𝑥^⋆

Definition 3.7. Let 𝜐 be an admissible ideal of subsets of {𝑥_𝑛;𝑛∈𝐾_𝑙}. Let {𝑛∈𝐾_𝑙;|𝑥_𝑛−𝑥^⋆|<𝜀}∉𝜐 be an infinite series of real numbers and let 𝜏 If 𝜏 is bounded, then {𝑛∈ℕ;|𝑥_𝑛−𝑥^⋆|<𝜀}∉𝜏 (𝑥^⋆) is said to be a {𝑥_𝑛;𝑛∈ℕ}.-𝜏 (ℕ-∑^∞𝑛=1𝑥_𝑛). If 𝑋={𝑥∈ℝ;𝑥isa𝜏-clusterpointof∑^∞𝑛=1𝑥_𝑛}. or 𝑋 is 𝑠=sup𝑋-cluster point (𝑠=inf𝑋 or 𝜏 is limsup_𝑛→∞∑^∞𝑛=1𝑥_𝑛-cluster point), then 𝜏-liminf_𝑛→∞∑^∞𝑛=1𝑥_𝑛 (sup𝑋=∞-∞).

Definition 3.8. We say that a sequence 𝜏 is inf𝑋=−∞-bounded above (−∞-bounded bellow), if there exist 𝜏 such that 𝜏 (limsup_𝑛→∞∑^∞𝑛=1𝑥_𝑛=∞) and is 𝜏-bounded if it is liminf_𝑛→∞∑^∞𝑛=1𝑥_𝑛=−∞-bounded above and below simultaneously.

It is obvious that if {𝑥_𝑛;𝑛∈ℝ} is faster convergent than 𝜏, then 𝜏 is 𝑚∈ℝ-faster convergent, the {𝑛∈ℕ;𝑥_𝑛>𝑚}∈𝜏. Generally, from the fact that {𝑛∈ℕ;𝑥_𝑛<𝑚}∈𝜏 is 𝜏-faster convergent, 𝜏 does not hold that ∑^∞𝑛=1𝑎_𝑛 is faster convergent than ∑^∞𝑛=1𝑏_𝑛.

It is obvious that for ∑^∞𝑛=1𝑎_𝑛-convergent series where 𝜏 has the property ∑^∞𝑛=1𝑏_𝑛:∑^∞𝑛=1𝑎_𝑛(𝜏 or ∑^∞𝑛=1𝑏_𝑛 asymptotic density ∑^∞𝑛=1𝑎_𝑛 have ∑^∞𝑛=1𝑏_𝑛), we obtain similar lemma as Lemma 2.6.

Lemma 3.9. Let 𝜏 be an admissible ideal of subsets of 𝜏 with property p1. Let if𝑀∈𝜏,then𝑀+1=𝑛+1;𝑛∈𝑀∩𝑁∈𝜏,𝑀−1=𝑛−1>0;𝑛∈𝑀∩𝑁∈𝜏(3.1), 𝜏_𝑓 be 𝜏_𝑠𝑡={𝐴⊂𝑁;-convergent real series with nonzero terms. Let (𝐴)=0} for all p1. Let
𝜏 Then (1)ℕ(2)p1(3)∑^∞𝑛=1𝑎_𝑛

Proof. First, we note that if ∑^∞𝑛=1𝑏_𝑛, then sequence 𝜏 is 𝑏_𝑛+𝑏_𝑛+1+…≠0,-bounded above (similarly for 𝑛∈ℕ). Next, we show that if 𝑡_𝑖(𝑎)=𝜏-liminf_𝑛→∞|||1+𝑎_𝑛+1𝑎_𝑛+𝑎^𝑛+2𝑎_𝑛+…|||,𝑡_𝑠(𝑎)=𝜏-limsup_𝑛→∞|||1+𝑎_𝑛+1𝑎_𝑛+𝑎^𝑛+2𝑎_𝑛𝑡+…_𝑖|||,(𝑏)=𝜏-liminf_𝑛→∞|||1+𝑏_𝑛+1𝑏_𝑛+𝑏^𝑛+2𝑏_𝑛+…|||,𝑡_𝑠(𝑏)=𝜏-limsup_𝑛→∞|||1+𝑏_𝑛+1𝑏_𝑛+𝑏^𝑛+2𝑏_𝑛+…|||.(3.2) is a if𝑡_𝑠(𝑎)<∞,𝑡_𝑖(𝑏)>0,𝜏-lim_𝑛→∞(𝑎_𝑛/𝑏_𝑛)=0,then𝜏-lim_𝑛→∞(𝑎_𝑛+𝑎_𝑛+1+…)/(𝑏_𝑛+𝑏_𝑛+1+-convergent sequence to 0 and …)=0, is a if𝑎_𝑛+𝑎_𝑛+1+…≠0forall𝑛∈ℕ,𝑡_𝑖(𝑎)>0,𝑡_𝑠(𝑏)<∞,𝜏-lim_𝑛→∞(𝑎_𝑛+𝑎_𝑛+1+…)/(𝑏_𝑛+-bounded sequence, then 𝑏_𝑛+1+…)=0,then𝜏-lim_𝑛→∞(𝑎_𝑛/𝑏_𝑛)=0, is a if𝑎_𝑛+𝑎_𝑛+1+…≠0forall𝑛∈ℕ,0<𝑡_𝑖(𝑎),𝑡_𝑠(𝑎)<∞,0<𝑡_𝑖(𝑏),𝑡_𝑠(𝑏)<∞,then-convergent sequence to 0. There exist 𝜏-lim_𝑛→∞(𝑎_𝑛/𝑏_𝑛)=0,ifandonlyif𝜏-lim_𝑛→∞(𝑎_𝑛+𝑎_𝑛+1+…)/(𝑏_𝑛+𝑏_𝑛+1+…)=0. such that the set 𝜏-limsup_𝑛→∞𝑥_𝑛<∞ is from {𝑥_𝑛;𝑛∈ℕ}. Let 𝜏 The set 𝜏-liminf is from {𝑎_𝑛;𝑛∈ℕ}. If 𝜏, then {𝑏_𝑛;𝑛∈ℕ} and 𝜏 and so {𝑎_𝑛𝑏_𝑛∶𝑛∈ℕ}. From this and from properties of ideal 𝜏, follows that 𝑘∈ℝ𝑁_𝑏={𝑛;|𝑏_𝑛|≥𝑘}-converges to 0. Then the proof follows from Lemma 2.6.

The following examples show that we cannot replace lim by 𝜏-lim in Lemmas 1.2, 2.1, and 2.2.

Example 3.10. Let 𝜀>0. be a convergent series with positive terms such that 𝑁_𝑎(𝜀)={𝑛;|𝑎_𝑛|≥𝜀/𝑘} is convergent (e.g., 𝜏, 𝑛∈ℕ⧵(𝑁_𝑎(𝜀)∪𝑁_𝑏)). Let |𝑎_𝑛|<𝜀/𝐾, |𝑏_𝑛|<𝑘 be an infinite subset of |𝑎_𝑛𝑏_𝑛|<𝜀, where 𝜏 is an arbitrary admissible ideal with property {𝑎_𝑛𝑏_𝑛∶𝑛∈ℕ} different from 𝜏. We define a real series 𝜏 in the following way:∑^∞𝑛=1𝑏_𝑛where ∑^∞𝑛=1(𝑏_𝑛+𝑏_𝑛+1+𝑏_𝑛+2+…) be a real number such that ∑^∞𝑛=1𝑏_𝑛=∑^∞𝑛=1𝛼^𝑛−1. We obtain 0<𝛼<1, 𝑀={𝑖_𝑗;1<𝑖_𝑗<𝑖_𝑗+1,𝑗∈ℕ} for all 𝑀∈𝜏.

Example 3.11. Let ℕ, 𝜏 be an infinite subset of p1, where 𝜏_𝑓 is an arbitrary admissible ideal with property ∑^∞𝑛=1𝑎_𝑛 different from 𝑎_𝑛=⎧⎪⎪⎨⎪⎪⎩1𝑛𝑏^𝑛𝑏,𝑛>1,𝑛≠𝑖_𝑗,𝑖_𝑗∈𝑀,𝑗=2,3,…_𝑖𝑗−1𝑐,+𝑏_{𝑖𝑗−1+1}+…𝑛𝑛∈=1𝑀,⧵𝑖₁,(3.3). Let 𝑐 be a sequence of positive real numbers such that ∑^∞𝑛=1𝑎_𝑛=∑^∞𝑛=1𝑏_𝑛𝜏-lim_𝑛→∞(𝑎_𝑛/𝑏_𝑛)=0 where (𝑎_𝑛+𝑎_𝑛+1+…)/(𝑏_𝑛+𝑏_𝑛+1+…)>1. Put 𝑛≥𝑖₂𝑀={𝑛_𝑗;𝑛₁=1,𝑛_𝑗<𝑛_𝑗+1,𝑛_𝑗+1−𝑛_𝑗≥2,𝑗∈ℕ}. Let 𝑀∈𝜏ℕ for 𝜏 be defined as follows: if p1, where 𝜏_𝑓 we put {𝐵_𝑛𝑗}^∞𝑗=1𝐵_𝑛𝑗+1<𝐵_𝑛𝑗<𝑚𝐵_𝑛𝑗+1,lim_𝑗→∞𝐵_𝑛𝑗=0, For 𝑚>1, we have 𝐴_𝑘=𝐵_𝑘/2^𝑘,, where 𝑘∈𝑀 From this and from the definition 𝐴_𝑘,𝐵_𝑘, it follows 𝑘∉𝑀 hence 𝑛_𝑗<𝑘<𝑛_𝑗+1-𝑗∈ℕ It is obvious that 𝜀_𝑛𝑗=(𝐴_𝑛𝑗−𝐴_𝑛𝑗+1)/(𝑛_𝑗+1−𝑛_𝑗), for 𝐴_𝑘=𝐴_𝑛𝑗−(𝑘−𝑛_𝑗)𝜀_𝑛𝑗,𝐵_𝑘=𝐵_𝑛𝑗−(𝑘−𝑛_𝑗)𝜀_𝑛𝑗. Let 𝑘∉𝑀 and 𝐴_𝑘/𝐵_𝑘<𝐴_𝑛𝑗/𝐵_𝑛𝑗+1=𝐵_𝑛𝑗/2^𝑛𝑗𝐵_𝑛𝑗+1<𝑚/2^𝑛𝑗 be real numbers. Put 𝑛_𝑗<𝑘<𝑛_𝑗+1.𝐴_𝑘,𝑘∈𝑀 The series lim_𝑘→∞(𝐴_𝑘/𝐵_𝑘)=0,𝜏 are convergent with positive terms and lim_𝑘→∞(𝐴_𝑘/𝐵_𝑘)=0.-(𝐴_𝑘−𝐴_𝑘+1)/(𝐵_𝑘−𝐵_𝑘+1)=1𝑘≠𝑛_𝑗−1,-𝑗>1.

Remark 3.12. If 𝐴₀>𝐴₁ have not the property 𝐵₀>𝐵₁, we get similar lemma as Lemma 3.9, but with shift indices of given series. (In general, it does not hold that if 𝑎_𝑛=𝐴_𝑛−1−𝐴_𝑛,-𝑏_𝑛=𝐵_𝑛−1−𝐵_𝑛,, then 𝑛∈ℕ.-∑^∞𝑛=1𝑎_𝑛,, and so on.)

4. Kummer Series

In [7], is showed: if ∑^∞𝑛=1𝑏_𝑛 is a convergent real series with positive terms and with an unknown sum 𝜏 and lim_𝑛→∞(𝑎_𝑛+𝑎_𝑛+1+…)/(𝑏_𝑛+𝑏_𝑛+1+…)=0, is a convergent real series with positive terms and with a known sum 𝜏, then if lim_𝑛→∞(𝑎_𝑛/𝑏_𝑛)=1. the Kummer series 𝜏, where p1 and 𝜏 for lim_𝑛→∞𝑎_𝑛=𝑎, has the same sum as 𝜏 and if lim_𝑛→∞𝑎_𝑛+1=𝑎 is a faster convergent series then ∑^∞𝑛=1𝑎_𝑛.

In the following lemma, we prove by using Lemma 2.6 the faster convergence of the Kummer series 𝑎 to the unknown sum of ∑^∞𝑛=1𝑐_𝑛 without conditions of positivity of 𝑐 and terms of lim_𝑛→∞(𝑎_𝑛/𝑐_𝑛)=𝑝>0∑^∞𝑛=1𝑏_𝑛𝑏₁=𝑎₁+𝑝(𝑐−𝑐₁)

Lemma 4.1. Let 𝑏_𝑛=𝑎_𝑛−𝑝𝑐_𝑛 be a convergent real series with nonzero terms and with the sum 𝑛≥2. Let ∑^∞𝑛=1𝑎_𝑛 Let 𝑏_𝑛>0 for all ∑^∞𝑛=1𝑎_𝑛. Let ∑^∞𝑛=1𝑏_𝑛 be a convergent series with nonzero terms and with the sum ∑^∞𝑛=1𝑎_𝑛. Let 𝑝. If for the series ∑^∞𝑛=1𝑎_𝑛,, where ∑^∞𝑛=1𝑏_𝑛, and ∑^∞𝑛=1𝑐_𝑛., for ∑^∞𝑛=1𝑎_𝑛, is 𝑎, for liminf_𝑛→∞|1+𝑎_𝑛+1/𝑎_𝑛+𝑎_𝑛+2/𝑎_𝑛+…|>0., and 𝑎_𝑛+𝑎_𝑛+1+…≠0, then𝑛∈ℕ and ∑^∞𝑛=1𝑐_𝑛 is a faster convergent series than 𝑐.

Proof. Since lim_𝑛→∞𝑎_𝑛/𝑐_𝑛=𝑝≠0, the proof follows from Lemma 2.6.

In the next example, we construct, by using of Remark 2.7 and Lemma 4.1, the Kummer series ∑^∞𝑛=1𝑏_𝑛 faster convergent than the series 𝑏₁=𝑎₁+𝑝(𝑐−𝑐₁) such that series 𝑏_𝑛=𝑎_𝑛−𝑝𝑐_𝑛𝑛≥2𝑏_𝑛≠0 does not have positive terms.

Example 4.2. Let 𝑛≥2 (the sum is unknown). It is evident that limsup_𝑛→∞|1+𝑏_𝑛+1/𝑏_𝑛+𝑏_𝑛+2/𝑏_𝑛+…|<∞ Let ∑^∞𝑛=1𝑏_𝑛=𝑎, then ∑^∞𝑛=1𝑏_𝑛 and from ∑^∞𝑛=1𝑎_𝑛, lim_𝑛→∞(𝑏_𝑛/𝑎_𝑛)=0, it follows ∑^∞𝑛=1𝑏_𝑛 (the sum is known). The Kummer series is ∑^∞𝑛=1𝑎_𝑛 where ∑^∞𝑛=1𝑎_𝑛,, ∑^∞𝑛=1𝑐_𝑛,. It is obvious that ∑^∞𝑛=1𝑏_𝑛 Because ∑^∞𝑛=1𝑎_𝑛=∑^∞𝑛=1(−1)^𝑛/(4^𝑛(3𝑛²+√𝑛)), is series with alternating signs such that for all limsup_𝑛→∞|𝑎_𝑛+1/𝑎_𝑛|=1/4<1/2. we have that ∑^∞𝑛=1𝑐_𝑛=∑^∞𝑛=1(−1)^𝑛/(4^𝑛𝑛²), we get for lim_𝑛→∞𝑎_𝑛/𝑐_𝑛=1/3 ∑^∞𝑛=1(𝑥^𝑛/𝑛²)=−∫^𝑥0(ln(1−𝑡)/𝑡)𝑑𝑡𝑥∈(−1,1) Hence ∑^∞𝑛=1(−1)^𝑛/(4^𝑛𝑛²)=∫^0−1/4(ln(1−𝑡)/𝑡)𝑑𝑡 is a faster convergent series than ∑^∞𝑛=1𝑏_𝑛=𝑏₁+∑^∞𝑛=2(𝑎_𝑛−(1/3)𝑐_𝑛), and it has the same sum.