Perhaps, the most astounding step in Ramanujan's proof of Betrand's postulate is his application of Stirling's approximation.
He starts with the following inequality:
$\log\Gamma(x) - 2\log\Gamma(\frac{1}{2}x + \frac{1}{2}) \le \log[x]! - 2\log[\frac{1}{2}x]! \le \log\Gamma(x+1) - 2\log\Gamma(\frac{1}{2}x + \frac{1}{2})$
Then, applying Stirling's approximation, Ramanujan gets to:
$\log[x]! - 2\log[\frac{1}{2}x]! < \frac{3}{4}x$ if $x > 0$
and
$\log[x]! - 2\log[\frac{1}{2}x]! > \frac{2}{3}x$ if $x > 300$
I would be very interested in understanding how Stirling's approximation gets us to these two conclusions.
As I understand it, Ramanujan is refering to Stirling's Approximation for the Gamma function which as I understand to be this (from Wikipedia):
$\Gamma(z) = \sqrt{\frac{2\pi}{z}}(\frac{z}{e})^{z}(1 + O(\frac{1}{z}))$
If someone could provide the details, I would greatly appreciate it! :-)