A new sequence related to the Euler–Mascheroni constant

In this paper, we provide a new quicker sequence convergent to the Euler–Mascheroni constant using an approximation of Padé type. Our sequence has a relatively simple form and higher speed of convergence. Moreover, we establish lower and upper bound estimates for the difference between the sequence and the Euler–Mascheroni constant.


Introduction
The Euler-Mascheroni constant γ = 0.5772156649015328 . . . is one of the most famous constants in analysis and number theory. It is the limit of the sequence γ n = 1 + 1 2 + · · · + 1 n -log n. (1.1) There are many famous problems related to the properties of this constant; for example, it is not known yet whether the Euler-Mascheroni constant is a rational number. In recent years, many researchers made great efforts in the area of concerning the rate of convergence of the sequence (γ n ) n≥1 and establishing sequences converging faster to the Euler-Mascheroni constant γ .
We begin with a brief overview of the relevant research.
In this paper, starting from the sequence (γ n ) n≥1 , we use an approximation of Padé type and provide a new convergent sequence for Euler-Mascheroni constant.
The Padé approximant is the best approximation of a function by a rational function and often gives better approximation of the function than truncating its Taylor series. For these reasons, Padé approximants are also used in computer calculations (see [3,30]).
Recall the Padé approximant of P(n) of order [1/2]: (1.13) We will use this Padé approximant p [1/2] (n) as an additional term to establish a new quicker sequence converging to the Euler-Mascheroni constant. More precisely, we consider the following sequence: (1.14) Furthermore, we will provide lower and upper bound estimates for the difference between the sequence and the Euler-Mascheroni constant.

Main results
Our main results are stated in the following theorem.

Theorem 2.1 Let
and let , Then we have the asymptotic expansion (2) as n → ∞, where B k are Bernoulli numbers. More explicitly, we have as n → ∞, where p = b 2 1 /(6b 1 -1). Furthermore, we have the following double inequality: Proof Using the representation of the harmonic sum in terms of digamma function (see [4]) and the asymptotic formula Hence (2) Using the power series expansion gives as n → ∞. Thus we obtain (2) From the assumption conditions a 1 = 1 24b 1 (1 -3b 1 ) , Note that, for all odd Bernoulli numbers B 2m-1 = 0 (m ≥ 2), the last expression can be rewritten as that is, (2) which is the desired Eq. (2.1) in Theorem 2.1.
On the other hand, from 1 9 ), which implies that b 1 and b 2 are the roots of the equation x 2 -6px + p = 0. Therefore, It is easy to observe that , and thus It follows from (2.6) that (2) which implies the desired Eq. (2.2) in Theorem 2.1.
Remark 3.3 For more results relating to the Euler constant, sequences, and some estimates, we refer the interested reader to Sîntǎmǎrian [23][24][25][26] and the references therein.

Conclusion
To provide a sequence converging faster to the Euler-Mascheroni constant, we construct a sequence (2) n by reference to the Padé approximant method, which improves the rate of convergence of the sequences introduced by Lu [16,18]. Our sequence depends on a real parameter and has a relatively simple form. It is worth noting that the method mentioned is also applicable to establishing estimates of bounds for some special means. For example, the method can be used for further study on the results obtained previously by Chu et al. [6][7][8][9], Qian and Chu [22], Yang et al. [31][32][33][34], and Zhao et al. [35].

Funding
The work of the first author is supported by the Natural Science Foundation of Fujian Province of China under Grant 2016J01023.