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An ultimate extremely accurate formula for approximation of the factorial function

Authors:

Institution:

(1) Department of Mathematics, Faculty of Sciences and Arts, Valahia University of Targovişte, 130082 Targovişte, Romania

Abstract:

We prove in this paper that for every x ≥ 0,

$\sqrt{2\pi e}\cdot e^{-\omega}\left( \frac{x+\omega}{e}\right) ^{x+\frac {1}{2}} < \Gamma(x+1)\leq\alpha\cdot\sqrt{2\pi e}\cdot e^{-\omega}\left( \frac{x+\omega}{e}\right)^{x+\frac{1}{2}}$

where ${\omega=(3-\sqrt{3})/6}$ and α = 1.072042464..., then

$\beta\cdot\sqrt{2\pi e}\cdot e^{-\zeta}\left(\frac{x+\zeta}{e}\right)^{x+\frac{1}{2}}\leq\Gamma(x+1) < \sqrt{2\pi e}\cdot e^{-\zeta}\left( \frac{x+\zeta}{e}\right)^{x+\frac{1}{2}},$

where ${\zeta=(3+\sqrt{3})/6}$ and β = 0.988503589... Besides the simplicity, our new formulas are very accurate, if we take into account that they are much stronger than Burnside’s formula, which is considered one of the best approximation formulas ever known having a simple form.

Keywords:

Mathematics Subject Classification (2000)" target="_blank">Mathematics Subject Classification (2000) Primary: 40A25 Secondary 26D07

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