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An asymptotic approximation of Wallis’ sequence

100%
Lampret V.
Open Mathematics
|
2012
|
tom 10
|
nr 2
775-787
EN
An asymptotic approximation of Wallis’ sequence W(n) = Πk=1n 4k 2/(4k 2 − 1) obtained on the base of Stirling’s factorial formula is presented. As a consequence, several accurate new estimates of Wallis’ ratios w(n) = Πk=1n(2k−1)/(2k) are given. Also, an asymptotic approximation of π in terms of Wallis’ sequence W(n) is obtained, together with several double inequalities such as, for example, $W(n) \cdot (a_n + b_n ) < \pi < W(n) \cdot (a_n + b'_n )$ with $a_n = 2 + \frac{1} {{2n + 1}} + \frac{2} {{3(2n + 1)^2 }} - \frac{1} {{3n(2n + 1)'}}b_n = \frac{2} {{33(n + 1)^{2'} }}b'_n \frac{1} {{13n^{2'} }}n \in \mathbb{N} $ .
2
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Local approximation properties of certain class of linear positive operators via I-convergence

64%
Özarslan M., Aktuǧlu H.
Open Mathematics
|
2008
|
tom 6
|
nr 2
281-286
EN
In this study, we obtain a local approximation theorems for a certain family of positive linear operators via I-convergence by using the first and the second modulus of continuities and the elements of Lipschitz class functions. We also give an example to show that the classical Korovkin Theory does not work but the theory works in I-convergence sense.
3
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I-convergence theorems for a class of k-positive linear operators

64%
Özarslan M.
Open Mathematics
|
2009
|
tom 7
|
nr 2
357-362
EN
In this paper, we obtain some approximation theorems for k- positive linear operators defined on the space of analytical functions on the unit disc, via I-convergence. Some concluding remarks which includes A-statistical convergence are also given.
4
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An accurate approximation of zeta-generalized-Euler-constant functions

52%
Lampret V.
Open Mathematics
|
2010
|
tom 8
|
nr 3
488-499
EN
Zeta-generalized-Euler-constant functions, $$ \gamma \left( s \right): = \sum\limits_{k = 1}^\infty {\left( {\frac{1} {{k^s }} - \int_k^{k + 1} {\frac{{dx}} {{x^s }}} } \right)} $$ and $$ \tilde \gamma \left( s \right): = \sum\limits_{k = 1}^\infty {\left( { - 1} \right)^{k + 1} \left( {\frac{1} {{k^s }} - \int_k^{k + 1} {\frac{{dx}} {{x^s }}} } \right)} $$ defined on the closed interval [0, ∞), where γ(1) is the Euler-Mascheroni constant and $$ \tilde \gamma $$(1) = ln $$ \frac{4} {\pi } $$, are studied and estimated with high accuracy.
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