Ramanujan's approximation to factorial I saw this approximation for the factorial given by Ramanujan as
$$\log(n!) \approx n \log n - n + \frac{\log(n(1+4n(1+2n)))}{6} + \frac{\log(\pi)}{2}$$ in wikipedia, which claims the approximation is superior to Stirling's approximation. I tried to locate the reference but unfortunately I could not.
I would appreciate if someone can throw light on how this asymptotic is obtained and the order of the error.
 A: Well, I finally found the formula in 
S. Ramanujan, The Lost Notebook and other Unpublished Papers. S. Raghavan and S. S. Rangachari, editors. Narosa, New Delhi, 1987.  
page 339. 
A: $$\eqalign{\log(n!) &\approx \left( \ln  \left( n \right) -1 \right) n+\ln  \left( \sqrt {2 \pi } \right) +\dfrac{\ln  \left( n \right)}{2} +\dfrac{1}{12} n^{-1}-{\dfrac {1}{
360}}\,{n}^{-3}+O \left( {n}^{-5} \right)\cr
n\ln  \left( n \right) &-n+\frac{\ln  \left( n \left( 1+4\,n \left( 1+2
\,n \right)  \right)  \right)}{6} +\frac{\ln  \left( \pi  \right)}{2}\cr &= \left( 
\ln  \left( n \right) -1 \right) n+\ln  \left( \sqrt {2 \pi } \right)  +\frac{
\ln  \left( n \right)}{2}  +\frac{1}{12}{n}^{-1}-{
\frac {1}{288}}\,{n}^{-3}+{\frac {1}{768}}\,{n}^{-4}+O \left( {n}^{-5}
 \right) \cr}
$$
So the error in Ramanujan's approximation is asymptotic to $\left(\dfrac{1}{288} - \dfrac{1}{360}\right) n^{-3} = \dfrac{1}{1440} n^{-3}$.
EDIT: an even better approximation, then, would be
$n\ln  \left( n \right) -n+\dfrac{\ln  \left(1/30 + n \left( 1+4\,n \left( 1+2
\,n \right)  \right)  \right)}{6} +\dfrac{\ln  \left( \pi  \right)}{2}$
where the error is asymptotic to $-\dfrac{11}{11520} n^{-4}$.  Thus at $n=10$
we have $\ln 10! \approx 15.1044125730755$, Ramanujan's approximation $\approx 
15.1044119983597$ and the improved approximation $\approx 15.1044126589476$.
