Improved convergence towards generalized Euler–Mascheroni constant

Abstract

We propose new simple sequences approximating the Euler–Mascheroni constant and its generalization, which converge faster towards their limits than those considered by DeTemple [D.W. DeTemple, A quicker convergence to Euler’s constant, Am. Math. Monthly 100 (5) (1998) 468–470], Sîntămărian [A. Sîntămărian, A generalization of Euler’s constant, Numer. Algorithms 46 (2) (2007) 141–151] and Vernescu [A. Vernescu, A new accelerate convergence to the constant of Euler, Gazeta Matem., Ser. A, Bucharest XVII(XCVI) (4) (1999) 273–278].

Introduction

One of the most important sequences in analysis and number theory of the form

$${\gamma }_n=1+\frac{1}{2}+\frac{1}{3}+\cdots +\frac{1}{n}-\ln n\text{,}$$

considered by Leonhard Euler in 1735, is known to converge towards the limit

$$\gamma =0.57721566490115328\dots \text{,}$$

which is now called the Euler–Mascheroni constant. It is known that this number can be approximated with a very high speed of convergence by the following series

$$\gamma =\frac{2^n}{e^{2^n}}\sum _{r=0}^{\infty }\frac{2^{\mathit{rn}}}{(r+1)!}\sum _{s=0}^r\frac{1}{s+1}-n\ln 2+O\left(\frac{1}{2^n{e}^{2^n}}\right)\text{,}$$

obtained by Bailey [3], who improved the earlier work of Sweeney [19]. Moreover, using the following equivalent definition of the Euler–Mascheroni constant

$$\gamma =\sum _{r=1}^{\infty }\left(\frac{1}{r}+\ln \frac{r}{r+1}\right)\text{,}$$

Alzer and Koumandos [2] established recently the following formula

$$\gamma =\sum _{r=1}^{\infty }\sum _{s=1}^r\left(\begin{array}{c}\hfill r-1\hfill \\ \hfill s-1\hfill \end{array}\right)(-1{)}^{r-s}{2}^s\left(\frac{1}{s}+\ln \frac{s}{s+1}\right)\text{,}$$

by using two series representations for $\psi (z)-\psi (y)$, where $\psi$ is the logarithmic derivative of the gamma function, $\psi ={\Gamma }^{\prime }/\Gamma$.

Several other formulas for the constant $\gamma$ can be found on the Wolfram Mathworld website. Evidently, all these expressions are too complicated to be suitable in numerical approximations of the constant $\gamma$. Hence other simplified methods were invented where, however, the speed of convergence is much slower than e.g. in the Bailey–Sweeney approximation.

First of all, we recall that the sequence $({\gamma }_n{)}_{n⩾1}$ converges very slowly to its limit, since

$$\frac{1}{2n+1}<{\gamma }_n-\gamma <\frac{1}{2n}\text{,}$$

(see [1], [8], [9], [22]). These estimates are not optimal because Tóth [20] proposed the following inequality

$$\frac{1}{2n+2/5}<{\gamma }_n-\gamma <\frac{1}{2n+1/3}\text{,}n⩾1$$

and Chen and Qi [4] proved that the constant 1/3 is best possible, however, 2/5 can be replaced by the sharp constant $(1-\gamma {)}^{-1}-2$. Next, Qiu and Vuorinen [17, Cor. 2.13] showed the double inequality

$$\frac{1}{2n}-\frac{\alpha }{n^2}<{\gamma }_n-\gamma ⩽\frac{1}{2n}-\frac{\beta }{n^2}\text{,}n⩾1\text{,}$$

where $\alpha =1/2$ and $\beta =\gamma -1/2$. Recall here that its direct consequence

$$\frac{1}{2n}-\frac{1}{8n^2}<{\gamma }_n-\gamma <\frac{1}{2n}$$

is called Franel’s inequality [16, Ex. 18]. Questions on the fast approximations of the Euler–Mascheroni constant $\gamma$ were also discussed by Karatsuba [6] and the following inequalities were obtained

$$\frac{1}{2n}-\frac{1}{12n^2}+\frac{1}{120n^4}-\frac{1}{126n^6}<{\gamma }_n-\gamma <\frac{1}{2n}-\frac{1}{12n^2}+\frac{1}{120n^4}\text{.}$$

Analogous questions were studied in the case of the generalized sequence

$${\gamma }_n(a)=\frac{1}{a}+\frac{1}{a+1}+\cdots +\frac{1}{a+n-1}-\ln \frac{a+n-1}{a}\text{,}\text{for}a>0$$

introduced in the monograph by Knopp [7]. Its limit, denoted by $\gamma (a)$, is also called the generalized Euler–Mascheroni constant, since $\gamma (1)=\gamma$.

As before, the sequence $({\gamma }_n(a){)}_{n⩾1}$ is known to converge slowly to its limit, since

$$\underset{n\to \infty }{\lim }n\left({\gamma }_n(a)-\gamma (a)\right)=\frac{1}{2}\text{,}$$

see Sîntămărian [18, Theorem 2.1].

Several authors have tried to estimate the speed of convergence of the sequence $({\gamma }_n(a){)}_{n⩾1}$. Here, one possible approach is to generalize DeTemple’s sequence

$$R_n=1+\frac{1}{2}+\frac{1}{3}+\cdots +\frac{1}{n}-\ln \left(n+\frac{1}{2}\right)\text{,}$$

which satisfies

$$\frac{1}{24(n+1{)}^2}<R_n-\gamma <\frac{1}{24n^2}\text{,}$$

see DeTemple [5].

Recently, Sîntămărian [18] has modified that argument by introducing the logarithmic term to the sequence $({\gamma }_n(a){)}_{n⩾1}$ to show that the new sequence

$${\lambda }_n(a)=\frac{1}{a}+\frac{1}{a+1}+\cdots +\frac{1}{a+n-1}-\ln \left(\frac{a+n-1}{a}+\frac{1}{2a}\right)\text{,}$$

has the faster convergence towards the limit $\gamma (a)$, since

$$\underset{n\to \infty }{\lim }{n}^2\left({\lambda }_n(a)-\gamma (a)\right)=\frac{1}{24}\text{,}$$

see [18, Theorem 3.1].

An analogous idea appears in the work by Vernescu [21], who replaced the last term in the series $({\gamma }_n{)}_{n⩾1}$ by its half, namely, he considered

$$V_n=1+\frac{1}{2}+\frac{1}{3}+\cdots +\frac{1}{n-1}+\frac{1}{2n}-\ln n$$

and showed

$$\frac{1}{12(n+1{)}^2}<\gamma -V_n<\frac{1}{12n^2}\text{.}$$

Notice that all attempts from the works [5], [18], [21] improved the speed of convergence from $n^{-1}$ to $n^{-2}\text{,}$ only.

In this work, we propose a combination of DeTemple’s sequence $(R_n{)}_{n⩾1}$ and Vernescu’s sequence $(V_n{)}_{n⩾2}$ which allows us to obtain a faster convergence of the sequence $({\gamma }_n(a){)}_{n⩾1}$ towards its limit. More precisely, we define the following sequence depending on two additional real parameters b and c, with $c>-(a+1)/a$, by the formula

$${\mu }_n(a\text{,}b\text{,}c)=\frac{1}{a}+\frac{1}{a+1}+\cdots +\frac{1}{a+n-2}+\frac{b}{a+n-1}-\ln \left(\frac{a+n-1}{a}+c\right)\text{.}$$

Notice that ${\mu }_n(a\text{,}1\text{,}0)={\gamma }_n(a)\text{,}{\mu }_n\left(a\text{,}1\text{,}\frac{1}{2a}\right)={\lambda }_n(a)\text{,}{\mu }_n(1\text{,}1\text{,}0)={\gamma }_n\text{,}{\mu }_n\left(1\text{,}1\text{,}\frac{1}{2}\right)=R_n\text{,}{\mu }_n\left(1\text{,}\frac{1}{2}\text{,}0\right)=V_n$.

Our main goal is to unify approaches of DeTemple and Vernescu by considering a single sequence, with the modified last term of the harmonic sum as well as with a new logarithmic term. Our main contribution is to find real parameters b and c for which we obtain a substantially faster convergence than the sequences considered in [5], [18], [21]. More precisely, we show that the sequences

$${\mu }_n\left(a\text{,}\frac{3+\sqrt{6}}{6}\text{,}\frac{1}{a\sqrt{6}}\right)\text{and}{\mu }_n\left(a\text{,}\frac{3-\sqrt{6}}{6}\text{,}\frac{-1}{a\sqrt{6}}\right)$$

converge towards their limit with speeds estimated by $n^{-3}$, as shown in Theorem 2.1. Furthermore, we show that the sequence

$${\rho }_n(a)=\frac{2}{a}+\frac{2}{a+1}+\cdots +\frac{2}{a+n-2}+\frac{1}{a+n-1}-\ln \left({\left(\frac{a+n-1}{a}\right)}^2-\frac{1}{6a^2}\right)$$

converges to its limit $2\gamma (a)$ with the speed of convergence bounded $n^{-4}\text{.}$