Editing Stirling's approximation (section)

==A convergent version of Stirling's formula==
[[Thomas Bayes]] showed, in a letter to [[John Canton]] published by the [[Royal Society]] in 1763, that Stirling's formula did not give a [[convergent series]].{{r|bayes-canton}}  Obtaining a convergent version of Stirling's formula entails evaluating [[Gamma function#Raabe's formula|Binet's formula]]:
<math display=block>\int_0^\infty \frac{2\arctan\left(\frac{t}{x}\right)}{e^{2\pi t}-1}\,{\rm d}t = \ln\Gamma(x) - x\ln x + x - \tfrac12\ln\frac{2\pi}{x}.</math>

One way to do this is by means of a convergent series of inverted [[rising factorial]]s. If
<math display=block>z^{\bar n} = z(z + 1) \cdots (z + n - 1),</math>
then
<math display=block>\int_0^\infty \frac{2\arctan\left(\frac{t}{x}\right)}{e^{2\pi t} - 1}\,{\rm d}t = \sum_{n=1}^\infty \frac{c_n}{(x + 1)^{\bar n}},</math>
where
<math display=block>c_n = \frac{1}{n} \int_0^1 x^{\bar n} \left(x - \tfrac{1}{2}\right)\,{\rm d}x = \frac{1}{2n}\sum_{k=1}^n \frac{k|s(n, k)|}{(k + 1)(k + 2)},</math>
where {{math|''s''(''n'',&nbsp;''k'')}} denotes the [[Stirling numbers of the first kind]]. From this one obtains a version of Stirling's series
<math display=block>\begin{align}
\ln\Gamma(x) &= x\ln x - x + \tfrac12\ln\frac{2\pi}{x} + \frac{1}{12(x+1)} + \frac{1}{12(x+1)(x+2)} + \\
  &\quad + \frac{59}{360(x+1)(x+2)(x+3)} + \frac{29}{60(x+1)(x+2)(x+3)(x+4)} + \cdots,
\end{align}</math>
which converges when {{math|Re(''x'') > 0}}. 
Stirling's formula may also be given in convergent form as<ref>{{cite book |last1=Artin |first1=Emil |title=The Gamma Function |date=2015 |page = 24|publisher=Dover }}</ref>
<math display=block>
\Gamma(x)=\sqrt{2\pi}x^{x-\frac{1}{2}}e^{-x+\mu(x)}
</math>
where 
<math display=block>
\mu\left(x\right)=\sum_{n=0}^{\infty}\left(\left(x+n+\frac{1}{2}\right)\ln\left(1+\frac{1}{x+n}\right)-1\right).
</math>