Abstract
The aim of this paper is to propose a new sequence that approximates the Euler–Mascheroni constant which converges faster towards its limit and to establish new inequalities for this constant.
1. Introduction
It is well known that the following sequence:is convergent to a limit denoted by , known as Euler–Mascheroni constant. Many authors have obtained different estimations for ; for example, the following inequalities increase better:
The convergence of the sequence to is very slow [1–3].
DeTemple [4] modified the logarithmic term of and showed that the following sequence:converges to with the rate of convergence since
Negoi [5] modified the logarithmic term of and showed that the following sequence:is strictly increasing and convergent to with the rate of convergence . Moreover, he proved that
Chen and Mortici [6] proved that for all integers , we havewith the best possible constants as
A convergence result to with the rates of convergence , respectively, was obtained by Mortici [7]:where , and
Yang [8] proved thatwhere .
Now, we define the sequence , for , and we prove that for all integers , we havewith the best possible constants .
2. The Main Result
Starting from the sequences and , we consider the family of sequencesfor , andwhich converge to zero.
Using a Maclaurin growth series, we get
If , then and so
It results that
Thus, we get
By a standard result, if a sequence converges to zero and there exists , then (see, e.g., [9]).
In our case of , we have and so
Starting from this result and using an elementary sequence method and MATLAB software for computation, we obtain the following:
Theorem 1. For every integer , we havewith the best possible constants .
Proof. We define the sequencefor and so whereThe derivative of function is equal toBy using MATLAB software, we obtain thatIf , thenfor all , and then is strictly decreasing.
We have and then it follows for all , such that is strictly increasing. Since converges to zero, it results that for all , such thatIf , then , for all and then is strictly increasing on .
Since , it results that for all , such that is strictly decreasing.
The sequence converges to zero and then it results that for all , such that
Remark 2. Let us remark that, if , then and then there exists such that for all and thenfor all .
Remark 3. Returning to the sequenceswith the rates of convergence ,, and , respectively, I used MATLAB software for computing the terms , and , with the first 10 exact decimals, for several iterations .
The data obtained are contained in Table 1, where we can see the faster convergence of the sequence to compared to and :
3. Conclusions
By modifying the logarithmic term of , we have constructed a new sequence that converges faster to the Euler–Mascheroni constant, with the convergence rate , compared to the sequence in [4] with convergence rate or those in [5, 6] with convergence rates .
Also, the idea of constructing the sequence allows the construction of a new sequence with the convergence rate , starting from the family of sequences:for .
More generally, starting from the family of sequences,for , we find a sequence with a convergence rate .
Data Availability
The data used to support the findings of this study are available from the author upon request.
Conflicts of Interest
The author declares that there are no conflicts of interest.
Acknowledgments
The APC was funded by “Dunarea de Jos” University of Galati, Romania.