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# A Refined Approximation of $$n!$$

Demonstrate, for $$n > 0$$, the bound $\sqrt { 2\pi n } { \left( \frac { 1 }{ e } \left( n+\frac { 1 }{ 12n } \right) \right) }^{ n } < n! < \sqrt { 2\pi n } { \left( \frac { n }{ e } \right) }^{ n }{ e }^{ \frac { 1 }{ 12n } }.$

I discovered this asymptotic bound of n factorial in 2010.

Solution

To see a full demonstration, you can read it at my blog.

Check out my other notes at Proof, Disproof, and Derivation

Note by Steven Zheng
2 years, 3 months ago