I'm reading Stigler's History of Statistics and am trying to understand as many of the derivations as I can. Stigler begins his discussion of DeMoivre's contributions by stating the result that the ratio of the maximum term of $(1+1)^n$ - that is, $\binom{n}{n/2}$ - to the sum of all terms, $2^n$, when $n$ is large is approximated by
$$ 2\frac{21}{125} \cdot \frac{\left(1-\frac{1}{n}\right)^n}{\sqrt{n-1}} $$
The constant, $A=2\frac{21}{125}=2.168$, Stigler says, was found by numerical evaluation of the first four terms of the series
$$ \ln\left(\frac{A}{2}\right)=\frac{1}{12}-\frac{1}{360}+\frac{1}{1260}-\frac{1}{1680}+\dots $$
As an aside, the constant $A$ was later shown by Stirling to be $\frac{2e}{\sqrt{2\pi}}$.
I'd like to understand DeMoivre's derivation of this result in detail, but so far my attempts to reproduce it have failed. I am, evidently, not DeMoivre's equal. Could someone please show me the way?
EDIT:
I was able to find this link which explains one possible line of argument. I am disturbed by the leap that leads to equation 1, but perhaps this was the norm in the 1700's?