Stirling’s Formula

Stirling’s formula says

where means that the ratio of the two quantities goes to as

Where does this formula come from? In particular, how does the number get involved? Where is the circle here?

To understand these things, I think a nonrigorous argument that can be made rigorous is more useful than a rigorous proof with all the ε’s dotted and the δ’s crossed. It’s important, I think, to keep the argument short. So let me do that.

The punchline will be that the comes from this formula:

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