Proyecciones Journal of Mathematics Vol. 31, N^o 1, pp. 29-38, March 2012. Universidad Católica del Norte Antofagasta - Chile

 

Improving some sequences convergent to Euler-Mascheroni constant

 

Necdet Batir

Nevsehir University, Nevsehir, Turkey

Chao-Ping Chen

Henan Polytechnic University, China

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ABSTRACT

We obtain the following very fast sequences convergent to Euler-Mascheroni constant:

= _H ( 1 J_ _J_ 23 17 10099 \

ⁿ = ⁿ־ ^ogVⁿ + 2 + 24n 48 ־n² + 5760n³ + 3840n⁴ 2903040 ־n⁵ )

and

= _H ( 1 1___37 10313 \

ö_ç = H_n ^lo^ⁿ + 2 + 24(n + é) 5760(n + 2)³ + 2903040(n + 2 f) '

where H_n are the harmonic numbers defined by H_n = 1+2 +¹+... + n·

subjclass [2000] : Primary 11Y60; Secondary 40A05.

Keywords : Euler-Mascheroni constant, harmonic numbers, in-equalities, asymptotic expansion.

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REFERENCES

[1] N. Batir, Sharp bounds for the psi function and harmonic numbers, Math. Inequal. Appl., No. 4, pp. 917-925, (2011).         [ Links ]

[2] C-P Chen, C. Mortici, New sequences converging towards the Euler-Mascheroni constant, Computer and Mathematics with Applications, doi:10.1016/j.camwa.2011.03.099, (2011).         [ Links ]

[3] C-P. Chen, Inequalities for the Euler-Mascheroni constant, Appl. Math. Lett., 23, pp. 161-164, (2010).

[4] C. Mortici, New approximation of the gamma function in terms of the digamma function, Appl. Math. Lett., 23, No. 1, pp. 97-100, (2010).         [ Links ]

[5] C. Mortici, Fast convergences toward Euler-Mascheroni constant, Comput. Appl. Math., 29, No. 3, pp. 479-491, (2010).         [ Links ]

[6] C. Mortici, On new sequences converging towards the Euler-Mascheroniconstant, Computer Math. Appl., 59, No. 8, pp. 2610-2614, (2010).         [ Links ]

[7] C. Mortici, Optimizing the rate of convergence of some new classes of sequences convergent to Euler constant, Analysis Appl., 8, No. 1, pp.99-107, (2010).         [ Links ]

[8] C. Mortici, A quicker convergence toward the constant with the logarithm term involving the constant e, Carpathian J. Math., 26, No. 1,pp. 86-91, (2010).         [ Links ]

[9] T. Negoi, A faster convergence to the constant of Euler, Gazeta Matematica, Seria A, 15, No. 94, pp. 113, (1997).         [ Links ]

[10] D. W. Temple, A geometric look at sequences that converge to Euler'sconstant, College Math. J., 37, pp. 128-131, (2006).         [ Links ]

[11] D. W. Temple, A quicker convergences to Euler's constant, Amer.Math. Monthly, 100 (5), pp. 468-470,(1993).         [ Links ]

[12] R. M. Young, Euler's constant, Math. Gaz., 75, pp. 187-190, (1991).         [ Links ]

Necdet Batir

Department of Mathematics, Faculty of Arts and Sciences, Nevsehir University, Nevsehir, Turkey

e-mail : nbatir@hotmail.com and

Chao-Ping Chen

School of Mathematics and Informatics, Henan Polytechnic University,

Jiaozuo City 454003,

Henan Province,

People's Republic of China

e-mail : chenchaoping@sohu.com

Received : November 2011. Accepted : December 2011