In my statistical physics course, I have frequently encountered the following approximation due to Stirling:

It’s very useful but my professor didn’t explain how good the approximation was. The derivation I found turns out to be very simple and so I can present it in a few lines here:

- Note that:
- Now, if we define

we have an upper-Riemann sum with . - So we basically have the following approximation:

- By the intermediate value theorem,where as defined previously.
- Let’s check how good this approximation is:
- This error grows very slowly. In fact, if i.e. the number of molecules in a glass of water, which is a minuscule error relative to the number of molecules.

Note: A pdf version of this blog post is available here.