![]() ![]() Esi, The generalized tripled difference of χ 3 sequence spaces, Global Journal of Mathematical Analysis, 3 (2), pp. Steinhaus Sur la convergence ordinaire et la convergence asymptotique, Colloq. Tripathy, Some I related properties of triple sequences, Selcuk J. Duden, Triple sequences and their statistical convergence, Selcuk J. Phu Rough convergence in normed linear spaces, Numer. Fast, Sur la convergence statistique, Colloq. Savas, On lacunary statistically convergent triple sequences in probabilistic normed space, Appl. Necdet Catalbas, Almost convergence of triple sequences, Global J. Definition : We say that a sequence (xn) converges if there exists x0 IR such that for every. It is found that for such sequences, convergence in a monotone norm (e.g., L,) on a, b to a continuous function implies uniform convergence of the sequence. Esi, On some triple almost lacunary sequence spaces defined by Orlicz functions, Research and Reviews: Discrete Mathematical Structures, 1 (2), pp. Let us now state the formal definition of convergence. Das, Some generalized triple sequence spaces of real numbers, J. Tripathy, Statistically convergent triple sequence spaces defined by Orlicz function, J Math. The list may or may not have an infinite number of terms in them although we will be dealing exclusively with infinite sequences in this class. ![]() Aytar Rough statistical Convergence, Numer. A sequence is nothing more than a list of numbers written in a specific order. Throughout the paper r be a nonnegative real number. Defining the set of rough statistical limit points of a triple sequence, we obtain rough statistical convergence criteria associated with this set. In this paper, we introduce the notion of rough statistical convergence of triple sequences. ![]() ![]() ( 8) extended the notion of rough convergence using the concept of ideals which automatically extends the earlier notions of rough convergence and rough statistical convergence. Aytar ( 1) extended the idea of rough convergence into rough statistical convergence using the notion of natural density just as usual convergence was extended to statistical convergence. In the mathematical field of real analysis, the monotone convergence theorem is any of a number of related theorems proving the convergence of monotonic sequences (sequences that are non-decreasing or non-increasing) that are also bounded. Sequences: definition of limit, proving results concerning limits of sequences, find- ing the limit of a bounded monotone sequence, proof and application of the. The idea of rough convergence occurs very naturally in numerical analysis and has interesting applications. The idea of rough convergence was introduced by Phu ( 9), who also introduced the concepts of rough limit points and roughness degree. The theory of statistical convergence has been discussed in trigonometric series, summability theory, measure theory, turnpike theory, approximation theory, fuzzy set theory and so on. Let K be a subset of the set N × N × N, and let us denote the set is different from zero, then ( xminjkℓ ) is called a non thin sub sequence of a triple sequence x.Ĭ ∈ R is called a statistical cluster point of a triple sequence x= ( xmnk ) provided that the natural density of the set The different types of notions of triple sequence was introduced and investigated at the initial by Sahiner et al. Statistical convergence is a generalization of the usual notion of convergence, which parallels the theory of ordinary convergence.Ī triple sequence (real or complex) can be defined as a function x: N × N × N → R( C), where N, R and C denote the set of natural numbers, real numbers and complex numbers respectively. use the algebra of limits of convergent sequences to compute the. If a n is a rational expression of the form, where P(n) and Q(n) represent polynomial expressions, and Q(n) ≠ 0, first determine the degree of P(n) and Q(n).The idea of statistical convergence was introduced by Steinhaus ( 12) and also independently by Fast ( 7) for real or complex sequences. define a Cauchy sequence, and apply the Cauchys Criterion for. Thus, the various methods used to find limits can also be applied when trying to determine whether a sequence converges. The figure below shows the graph of the first 25 terms of the sequence, which demonstrates the trend of the sequence towards 2 (though alone it would not be sufficient to conclude that the sequence converges to 2).Ī sequence converges if the limit of its nth term exists and is finite. ![]()
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