# How accurate is Stirling’s approximation?

The question whether an approximation is good should not be asked in isolation, because the answer depends on the purpose. For example, $$\mathrm{e}^x \approx 1 + x$$ works well for small (positive) $$x$$. How small? If we want to keep the relative error below 10%, $$x \lt 0.5$$ will do fine. But if we need more precision, the range gets narrower, e.g., $$x \lt 0.15$$ if the tolerance is smaller than 1%.

But Stirling’s formula—which gives an approximate value for factorials: $$n! \sim \sqrt{2 \pi n} \left( \frac{n}{\mathrm{e}} \right)^n$$ for $$n \gg 1$$—works well even for small integers.

For $$x = 10$$ the relative error is less than 1%, for $$x = 100$$ the relative error is less than 0.1%, and so on.