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I recently came across the approximation $$\ln n! \approx n\ln n - n + \frac{1}{6}\ln(n(1+4n(1+2n)))+\frac{1}{2}\ln \pi$$ Supposedly this was written in Ramanujan's last notebook, but I can't seem to find it anywhere else, nevermind a proof for it. Does anyone know if it was ever proven, and if so where such a proof is?

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marked as duplicate by Vedvart1, user228113, J. M. is a poor mathematician, user91500, Claude Leibovici Sep 3 '17 at 9:01

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    $\begingroup$ Sorry to say this, but it seems like this question has already been asked by someone else on Math SE. $\endgroup$ – Toby Mak Sep 3 '17 at 4:12
  • $\begingroup$ Thanks for the link, not sure why I didn't come across that earlier. $\endgroup$ – Vedvart1 Sep 3 '17 at 4:19

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