Multiple-correction and faster approximation

Abstract

In this paper, we formulate a new multiple-correction method. The goal is to accelerate the rate of convergence. In particular, we construct some sequences to approximate the Euler-Mascheroni and Landau constants, which are faster than the classical approximations in literature.

Multiple-correction and continued fraction approximation (II)

“…The multiple-correction method First, let us briefly review a so-called multiple-correction method presented in our previous paper [6]. Let (v(n)) n≥1 be a sequence to be approximated.…”

“…(5.31) = (n − 2) ϑ −1 n−2 + ϑ 0 + ϑ 1 n + · · · + ϑ 11 n 11 3810240000π 2 (−12 + π 2 )(1 + n) 7 (3 + 2n) 6 (13 + 8n) 6 , and all coefficients ϑ j (−1 ≤ j ≤ 11) are negative. Thus, we obtain It is also known as the Euler-Mascheroni constant.…”

“…The main aim of this paper is to further develop a multiple-correction method formulated in a previous work [6]. As its applications, we find a kind of hybrid-type finite continued fraction approximations in two cases of Landau constants and Lebesgue constants.…”

“…Recently, Chen [10] found the following better approximation for G(n): as n → ∞, More recently, Cao, Xu and You [6] improved the rate of convergence to n −14 , and attained the following tight double-sides inequalities C 1 (n + 3 2 ) 6 < G(n) − 1 π ln(n + 3 4 ) − c 0 −…”

Some new quicker convergences to Glaisher–Kinkelin’s and Bendersky–Adamchik’s constants

“…Motivated by the important work of Mortici [2] and Lu [1], in this paper we will continue our previous works [7][8][9][10], and apply the multiple-correction method to construct some new convergent sequences for Glaisher-Kinkelin's and BenderskyAdamchik's constants, which have faster rate of convergence. Moreover, we establish sharp bounds for the corresponding error terms.…”