Stirling's approximation for $n!$

Stirling's approximation says that factorials can be approximated as follows: $$ n! \approx \sqrt{2\pi} \; n^{n+1/2} \; e^{-n}. $$ This is useful in statistical physics, for example, where we have to think about $n!$ for very large numbers $n$. It is also theoretically interesting.

In this notebook (which covers roughly the same ground as this video) we will simply plot some graphs to show that the approximation looks plausible.

import math
import numpy as np
import matplotlib.pyplot as plt

def stirling(n):
    """Return Stirling's approximation for n!"""
    return np.sqrt(2*np.pi)* n**(n+1/2)*np.exp(-n)

Because the numbers $n!$ grow so quickly, if we just plot them directly we get a picture that is very hard to interpret. It is much better use a logarithmic scale on the $y$-axis, which we do by including the line plt.yscale('log'). (Try removing that line to see what happens.)

ns = range(1, 21)
xs = np.linspace(1, 20, 100)
plt.xlim(0, 21)
plt.xticks(range(0, 21, 2))
plt.yscale('log')
plt.plot(xs, stirling(xs), 'r-',label='Stirling')
plt.plot(ns, [math.factorial(n) for n in ns], 'b.', label='n!')
plt.legend()

<matplotlib.legend.Legend at 0x20f25600510>