Stirling's Formula/Refinement/Examples/8

Example of Use of the Refinement to Stirling's Formula

$8! \approx 40 \, 318$

which shows an error of about $0.005 \%$.

$\ds n!$

$\approx$

$\ds \sqrt {2 \pi n} \paren {\dfrac n e}^n \paren {1 + \dfrac 1 {12 n} }$

$\ds \leadsto \ \ $

$\ds 8!$

$\approx$

$\ds \sqrt {2 \pi 8} \paren {\dfrac 8 e}^8 \paren {1 + \dfrac 1 {12 \times 8} }$

$\ds $

$=$

$\ds 4 \sqrt \pi \paren {\dfrac 8 e}^8 \paren {1 + \dfrac 1 {96} }$

$\ds $

$=$

$\ds 2^{26} \sqrt \pi e^{-8} \dfrac {97} {96}$

$\ds $

$=$

$\ds 67 \, 108 \, 864 \times 1.77245 \times 0.00033 \, 546 \times 1.01014$

$\ds $

$\approx$

$\ds 40 \, 318$

We have that by the usual calculation:

$8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40 \ 320$

Hence:

$\ds \frac {40320 - 40318} {40320}$

$=$

$\ds \frac 2 {40320}$

$\ds $

$\approx$

$\ds 0.00005$

$\ds $

$\approx$

$\ds 0.005 \%$

$\blacksquare$