# Stirling's formula

Stirling showed that with the constant $k = e$ the sequence $(x_{n})$ with $x_{n} = n! \Large\frac{k^{n}}{n^{n+\frac 1 2}}$ converges to $\sqrt{2\pi}$.

This means that for large $n$ we have the approximation $n! \approx \sqrt {2\pi n} \Large (\frac{n}{e})\large ^{n}$.

 n n! Stirling's approximation 10 3.629 × 106 3.604 × 106 100 9.333 × 10157 9.425 × 10157 1000 4.024 × 102576 4.464 × 102576