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### What is the asymptotic “inverse” of the factorial? [duplicate]

Take the following algorithm ...
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### How to solve equations with Stirling Approximation? [duplicate]

As we all know Stirling Approximation is giving us an approximate value of factorial, aka $\Gamma(x + 1)$. $\sqrt{2\pi n}(\frac{n}{e}) \approx n!$ But what if we have equations with factorials. In ...
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### On Ramanujan's approximation, $n!\sim \sqrt{\pi}\big(\frac ne\big)^n\sqrt [6]{(2n)^3+(2n)^2+n+\frac 1{30}}$

Over here I discovered that Ramanujan gave the following factorial approximation, better than Stirling's formula: $$n!\sim \sqrt{\pi}\left(\frac ne\right)^n\sqrt [6]{(2n)^3+(2n)^2+n+\frac 1{30}}$$ ...