Stirling's Formula/Examples/8

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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 $1 \%$.


Proof

\(\ds n!\) \(\approx\) \(\ds \sqrt {2 \pi n} \left({\dfrac n e}\right)^n\) Stirling's Formula
\(\ds \leadsto \ \ \) \(\ds 8!\) \(\approx\) \(\ds \sqrt {2 \pi 8} \left({\dfrac 8 e}\right)^8\)
\(\ds \) \(=\) \(\ds 4 \sqrt \pi \left({\dfrac 8 e}\right)^8\)
\(\ds \) \(=\) \(\ds 2^{26} \sqrt \pi e^{-8}\)
\(\ds \) \(=\) \(\ds 67108864 \times 1.77245 \times 0.00033546\)
\(\ds \) \(\approx\) \(\ds 39902\)


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:

\(\ds \frac {40320 - 39902} {40320}\) \(=\) \(\ds \frac {418} {40320}\)
\(\ds \) \(\approx\) \(\ds 0.01036\)
\(\ds \) \(\approx\) \(\ds 1.03 \%\)

$\blacksquare$


Sources