An approximation formula for n!

Authors

  • Necdet Batir Nevsehir Haci Bektas Veli Üniversitesi.

DOI:

https://doi.org/10.4067/S0716-09172013000200006

Keywords:

Gamma function, Stirling formula, Euler-Mascheroni constant, Harmonic numbers, Inequalities, Digamma function.

Abstract

We prove the following very accurate approximation formula for the factorial function:

n!p ???-???(? + 1+ 72(3(¾¾¾!+!)2332800 -(^ +1? This gives better results than the following approximation formula

, at- n -n I 1 1 31 139 9871

?! Pá V27rnne n\ n +---1--------H---,

V 6 72n 6480n2 155520?3 6531840?4'

which is established by the author [5] and C. Mortici [16] independently, and gives similar results with

32 32 ? n 176 128

, r- (?\n8/???? 32176 ~? ?! Pá ?/? — \ 16?4 + — ?3 + — ?2+ —— ? Ve/ V 3 9 405

3 9 405 1215

which is established by C. Mortici in his very new paper [8].

References

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[9] C. Mortici, Ramanujan’s estimate for the gamma function via monotonicity arguments, Ramanujan J, DOI.10.1007/s11139-010-9265-y, (2011).

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[11] C. Mortici, Accurate estimates of the gamma function involving the psi function, Numer. Funct. Anal. Optim., 32, no. 4, pp. 469-476, (2011).

[12] C. Mortici, New sharp inequalities for approximating the factorial function and the digamma function, Miskolc Math., 11, no. 1, pp. 79-86, (2010).

[13] C. Mortici, Asymptotic expansions of the generalized Stirling approximation, Mathe. Comput. Model., 52, no. 9-10, pp. 1867-1868, (2010).

[14] C. Mortici, Estimating gamma function in terms of digamma function, Math. Comput. Model., 52, no. 5-6, pp. 942-946, (2010).

[15] C. Mortici, On the Stirling expansion into negative powers of a triangular numbers, Math. Commun., 15, no. 2, pp. 359-364, (2010).

[16] C. Mortici, Sharp inequalities related to Gosper’s Formula, Comptes Rendus Mathematique, 348, no. 3-4, pp. 137-140, (2010).

[17] C. Mortici, A class of integral approximations for the factorial function, Computers and Mathematics with Applications, 59, no. 6, pp. 2053- 2058, (2010).

[18] C. Mortici, New improvements of the Stirling formula , Appl. Math. Comput., 217, no. 2, pp. 699-704, (2010).

[19] C. Mortici, New approximations of the gamma function in terms of the digamma function, Applied Mathematics Letters, 23, no. 1, pp. 97-100, (2010).

[20] C. Mortici, An ultimate extremely accurate formula for approximation of the factorial function, Arch. Math. (Basel), 93, no. 1, pp. 37-45, (2009).

How to Cite

[1]
N. Batir, “An approximation formula for n!”, Proyecciones (Antofagasta, On line), vol. 32, no. 2, pp. 173-181, 1.

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Artículos