Some new continued fraction approximation of Euler's constant

Abstract

In this paper, using continued fraction, some quicker classes of sequences convergent to Euler's constant are provided. Finally, for demonstrating the superiority of our new convergent sequences over DeTemple's sequence, Vernescu's sequence and Mortici's sequences, some numerical computations are also given.

Acceleration Methods for Series: A Probabilistic Perspective

“…Using continued fractions, Lu et al [23,24] have obtained monotone convergence to γ 0 of the order of O(n −p ), p = 3, 4, 5 (see also Lu [22] and Yang [25]). Kh.…”

A series of sequences convergent to Euler’s constant

“…They proved that, among the sequences , in the case of and the privileged sequence offers the best approximations of γ since Recently, Lu, Song, and Yu [10] provided some approximations of Euler’s constant. A new important sequence was defined as follows: where Two particular sequences were provided as These two sequences converge faster than all other sequences mentioned since for all , …”

Monotone and fast computation of Euler’s constant

“…With respect to the first type of computations, we mention that Xu and You [ 2 ] and Lu et al [ 3 , 4 ] have used continued fractions to obtain monotone convergence to γ . For instance, it is shown in [ 2 ], Theorem 2, that where C and are explicit constants, and and are sequences involving the logarithm of a continued fraction, the first one being strictly increasing, and the second one strictly decreasing.…”