Improved convergence towards generalized Euler–Mascheroni constant
Introduction
One of the most important sequences in analysis and number theory of the formconsidered by Leonhard Euler in 1735, is known to converge towards the limitwhich 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 seriesobtained by Bailey [3], who improved the earlier work of Sweeney [19]. Moreover, using the following equivalent definition of the Euler–Mascheroni constantAlzer and Koumandos [2] established recently the following formulaby using two series representations for , where is the logarithmic derivative of the gamma function, .
Several other formulas for the constant can be found on the Wolfram Mathworld website. Evidently, all these expressions are too complicated to be suitable in numerical approximations of the constant . 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 converges very slowly to its limit, since(see [1], [8], [9], [22]). These estimates are not optimal because Tóth [20] proposed the following inequalityand Chen and Qi [4] proved that the constant 1/3 is best possible, however, 2/5 can be replaced by the sharp constant . Next, Qiu and Vuorinen [17, Cor. 2.13] showed the double inequalitywhere and . Recall here that its direct consequenceis called Franel’s inequality [16, Ex. 18]. Questions on the fast approximations of the Euler–Mascheroni constant were also discussed by Karatsuba [6] and the following inequalities were obtainedAnalogous questions were studied in the case of the generalized sequenceintroduced in the monograph by Knopp [7]. Its limit, denoted by , is also called the generalized Euler–Mascheroni constant, since .
As before, the sequence is known to converge slowly to its limit, sincesee Sîntămărian [18, Theorem 2.1].
Several authors have tried to estimate the speed of convergence of the sequence . Here, one possible approach is to generalize DeTemple’s sequencewhich satisfiessee DeTemple [5].
Recently, Sîntămărian [18] has modified that argument by introducing the logarithmic term to the sequence to show that the new sequencehas the faster convergence towards the limit , sincesee [18, Theorem 3.1].
An analogous idea appears in the work by Vernescu [21], who replaced the last term in the series by its half, namely, he consideredand showedNotice that all attempts from the works [5], [18], [21] improved the speed of convergence from to only.
In this work, we propose a combination of DeTemple’s sequence and Vernescu’s sequence which allows us to obtain a faster convergence of the sequence towards its limit. More precisely, we define the following sequence depending on two additional real parameters b and c, with , by the formulaNotice that .
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 sequencesconverge towards their limit with speeds estimated by , as shown in Theorem 2.1. Furthermore, we show that the sequenceconverges to its limit with the speed of convergence bounded
Section snippets
Main result
Our approximating sequences of the generalized Euler–Mascheroni constant are gathered in the following theorem. Theorem 2.1 Let be given and satisfy and . If , the speed of convergence of the sequence defined in (1.1) is equal to , since If and , the speed of convergence of the sequenceequals , since If and , the
Concluding remarks
It is proved in [18] that the arithmetic mean of the sequencesandhas the speed of convergence equal to , although the sequences and converge to as , only. By the method presented in this work, we obtain other sequences with faster convergence. Indeed, adding the asymptotic relations (2.8), (2.9) we obtainwhere takes the form
References (22)
New approximations of the gamma function in terms of the digamma function
Appl. Math. Lett.
(2010)Inequalities for the gamma and polygamma functions
Abh. Math. Sem. Univ. Hamb.
(1998)- et al.
Series representations for and other mathematical constants
Anal. Math.
(2008) Numerical results on the transcendence of constants involving , and Euler’s constant
Math. Comput.
(1988)- et al.
The best lower and upper bounds of harmonic sequence
RGMIA
(2003) A quicker convergence to Euler’s constant
Am. Math. Monthly
(1998)On the computation of the Euler constant . Computational methods from rational approximation theory (Wilrijk, 1999)
Numer. Algorithms
(2000)- (1951)
- et al.
An improvement of the convergence speed of the sequence converging to Euler’s constant
An. Şt. Univ. Ovidius Constanţa.
(2005) - et al.
Some new facts in discrete asymptotic analysis
Math. Balkanica, New Ser.
(2007)
Cited by (56)
Multiple-correction and continued fraction approximation
2015, Journal of Mathematical Analysis and ApplicationsOptimal rate of convergence for sequences of a prescribed form
2013, Journal of Mathematical Analysis and ApplicationsCitation Excerpt :Recently, variations of the Euler–Mascheroni sequence with an improved order of convergence have been analyzed by Chen [6], Mortici [23–26] and Sîntămărian [36,37].
Limits and inequalities associated with the Euler-Mascheroni constant
2013, Applied Mathematics and ComputationFurther improvements of some double inequalities for bounding the gamma function
2013, Mathematical and Computer ModellingCitation Excerpt :In the past, especially in recent decades, several authors proved many interesting inequalities for the Euler gamma function (we refer the reader to [1–21] and all the references given therein).
A solution to an open problem on the Euler-Mascheroni constant
2013, Applied Mathematics and ComputationCitation Excerpt :Euler–Mascheroni-type sequences with an improved order of convergence were also considered by Chen [1], Mortici [10–13] and Sıˆntămărian [15–17].
Effective edge node configuration for video transport over Optical Burst Switched networks
2013, Optical Switching and Networking