I was reading about Stirling formula and then I saw the famous Stirling series which is as follows: $$n!\sim\sqrt{2\pi n}\left(\frac{n}{e}\right)^{n}\left(1+\frac{1}{12n}+\frac{1}{288n^{n}}+...\right)$$ The coefficient used in this series are called Stirling coefficients, there exist an explicit formula for calculating them which was given by G. Nemes.
The explicit formula is complicated and I prevent of mentioning that, the questions are :
Which one of Stirling series or Stirling formula has been discovered first?
Can someone explain what is the idea behind defining such a series? where does actually this come from? (This is indeed the most important one, I even can compute the coefficients, but have no idea about the notion of Stirling series)
Also Wikipedia says the Stirling formula is in fact the first approximation to the following series, what does that mean?how we can reach Stirling formula using this series?