Abstract
In this paper, we give some new quicker convergent sequences toward Euler’s constant using the continued fraction. For demonstrating the superiority of the new sequences over DeTemple’s sequence, Mortici’s sequences and Lu’s sequences, some numerical simulations are also given in this article.
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References
DeTemple, D.W.: A quicker convergences to Euler’s constant. Am. Math. Mon. 100(5), 468–470 (1993)
DeTemple, D.W.: A gemetric look at sequences that converge to Euler’s constant. College Math. J. 37, 128–131 (2006)
Lu, D.: A new quicker sequence convergent to Euler’s constant. J. Number Theory 136, 320–329 (2014)
Lu, D.: Some quicker classes of sequences convergent to Euler’s constant. Appl. Math. Comput. 232, 172–177 (2014)
Lu, D.: Some new convergent sequences and inequalities of Euler’s constant. J. Math. Anal. Appl. 419(1), 541–552 (2014)
Lu, D., Song, L., Yu, Y.: Some new continued fraction approximation of Euler’s constant. J. Number Theory 147, 69–80 (2015)
Mortici, C., Vernescu, A.: An improvement of the convergence speed of the sequence \((\gamma _n)_{n\ge 1}\) converging to Euler’s constant. An. Ştiinţ. Univ. Ovidius Constanţa Ser. Mat. 13(1), 97–100 (2005)
Mortici, C., Vernescu, A.: Some new facts in discrete asymptotic analysis. Math. Balk. (N.S.) 21(3–4), 301–308 (2007)
Mortici, C.: Optimizing the rate of convergence in some new calsses of sequences convergent to Euler’s constant. Anal. Appl. (Singap.) 8(1), 99–107 (2010)
Mortici, C.: Very accurate estimates of the polygamma functions. Asymptot. Anal. 68(3), 125–134 (2010)
Mortici, C.: A quicker convergence toward the gamma constant with the logarithm term involving the constant e. Carpath. J. Math. 26(1), 86–91 (2010)
Mortici, C.: On new sequences converging towards the Euler–Mascheroni constant. Comput. Math. Appl. 59(8), 2610–2614 (2010)
Mortici, C.: Product approximations via asymptotic integration. Am. Math. Mon. 117(5), 434–441 (2010)
Mortici, C.: A new Stirling series as continued fraction. Numer. Algorithms 56(1), 17–26 (2011)
Mortici, C.: A continued fraction approximation of the gamma function. J. Math. Anal. Appl. 402(2), 405–410 (2013)
Vernescu, A.: A new accelerate convergence to the constant of Euler. Gaz. Matem. Ser. A Buchar. 104(4), 273–278 (1999). (in Romanian)
Vernescu, A.: The order of convergence of the sequence that defines the constant of Euler. Gaz. Matem. Ser. A Buchar. 88(10–11), 380–381 (1983). (in Romanian)
Young, R.M.: Euler’s constant. Math. Gaz. 75(472), 187–190 (1991)
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Hu, X., Lu, D. & Wang, X. New Quicker Sequences and Inequalities with Continued Fraction Towards Euler’s Constant. Results Math 73, 28 (2018). https://doi.org/10.1007/s00025-018-0796-7
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DOI: https://doi.org/10.1007/s00025-018-0796-7