Cowboy mathematics

As a physicist, one often disregards the formalities of mathematics which can hinder ones investigations just as often as they can help. In this section, we present an informal rough derivation of Stirling's formula. Consider that

$$\ln(n!)=\sum_{k=1}^{n}\ln(k).$$ (1.1)

Multiply both sides by $\Delta k$, so,

$$\lim_{\Delta k \rightarrow 0}\sum_{k=1}^{n}\ln(k)\Delta k = \int \ln(k)dk = k\ln(k)-k$$ (1.2)

Now of course, in our sum $\Delta k=1$, but the relative size of $\Delta k$ becomes smaller as $n\rightarrow \infty$, so if we write,and so we find loose enough reason to approximate, for $n \gg 1$,

$$\ln(n!)\approx n\ln(n)-n$$ (1.3)