Stirling's Formula/Examples/8

Example of Use of Stirling's Formula

The factorial of $8$ is given by Stirling's Formula as:

$8! \approx 39 \ 902$

which shows an error of about $1 \%$.

$\ds n!$

$\approx$

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

$\ds \leadsto \ \ $

$\ds 8!$

$\approx$

$\ds \sqrt {2 \pi 8} \paren {\dfrac 8 e}^8$

$\ds $

$=$

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

$\ds $

$=$

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

$\ds $

$=$

$\ds 67108864 \times 1.77245 \times 0.00033546$

$\ds $

$\approx$

$\ds 39902$

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:

$\ds \frac {40320 - 39902} {40320}$

$=$

$\ds \frac {418} {40320}$

$\ds $

$\approx$

$\ds 0.01036$

$\ds $

$\approx$

$\ds 1.03 \%$

$\blacksquare$

1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.5$: Permutations and Factorials