A new sequence convergent to Euler–Mascheroni constant

In this paper, we provide a new sequence converging to the Euler–Mascheroni constant. Finally, we establish some inequalities for the Euler–Mascheroni constant by the new sequence.


Introduction
The Euler-Mascheroni constant was first introduced by Leonhard Euler (1707-1783) in 1734 as the limit of the sequence There are many famous unsolved problems about the nature of this constant (see, e.g., the survey papers or books of Brent and Zimmermann [1], Dence and Dence [2], Havil [3], and Lagarias [4]). For example, it is a long-standing open problem if the Euler-Mascheroni constant is a rational number. A good part of its mystery comes from the fact that the known algorithms converging to γ are not very fast, at least when they are compared to similar algorithms for π and e. The sequence (γ (n)) n∈N converges very slowly toward γ , like (2n) -1 . Up to now, many authors are preoccupied to improve its rate of convergence; see, for example, [2,[5][6][7][8][9][10][11][12][13][14] and references therein. We list some main results: Hence the rate of the convergence of the sequence (ν(n)) n∈N is n -12 . Very recently, by inserting the continued fraction term into (1.1), Lu [9] introduced a class of sequences (r k (n)) n∈N (see Theorem 1) and showed that 1 120(n + 1) 4 < r 3 (n)γ < 1 120(n -1) 4 . (1.4) In fact, Lu [9] also found a 4 without proof, and his works motivate our study. In this paper, starting from the well-known sequence γ n , based on the early works of Mortici, DeTemple, and Lu, we provide some new classes of convergent sequences for the Euler-Mascheroni constant. , Furthermore, for r 2 (n) and r 3 (n), we also have the following inequalities.
Theorem 2 Let r 2 (n) and r 3 (n) be as in Theorem 1. Then (1.7) Remark 1 Certainly, there are similar inequalities for r k (n) (1 ≤ k ≤ 7); we omit the details.

Proof of Theorem 1
The following lemma gives a method for measuring the rate of convergence. This lemma was first used by Mortici [15,16] for constructing asymptotic expansions or accelerating some convergences. For a proof and other details, see, for example, [16]. We need to find the value a 1 ∈ R that produces the most accurate approximation of the form To measure the accuracy of this approximation, we usually say that an approximation (2.3) is better as r 1 (n)γ faster converges to zero. Clearly, Developing expression (2.4) into a power series expansion in 1/n, we obtain From Lemma 1 we see that the rate of convergence of the sequence (r 1 (n)γ ) n∈N is even higher as the value s satisfies (2.1). By Lemma 1 we have (i) If a 1 = 1/2, then the rate of convergence of the (r 1 (n)γ ) n∈N is n -2 , since lim n→∞ n r 1 (n)γ = 1 2 a 1 = 0.
We also observe that the fastest possible sequence (r 1 (n)) n∈N is obtained only for a 1 = 1/2.
We repeat our approach to determine a 1 to a 7 step by step. In fact, we can easily compute a k , k ≤ 15, by the Mathematica software. In this paper, we use the Mathematica software to manipulate symbolic computations. Let Hence the key step is to expand r k (n)r k (n + 1) into power series in 1/n. Here we use some examples to explain our method.
We can use this approach to find a k (1 ≤ k ≤ 15). From the computations we may the conjecture a n+1 = C n , n ≥ 1. Now, let us check it carefully.
Remark 2 From the computations we can guess that a n+1 = C n , n ≥ 1. It is a very interesting problem to prove this. However, it seems impossible by the provided method.

Proof of Theorem 2
Before we prove Theorem 2, let us give a simple inequality, which follows from the Hermite-Hadamard inequality and plays an important role in the proof. By P k (x) we denote polynomials of degree k in x such that all its nonzero coefficients are positive; it may be different at each occurrence.