Only Stirling?

Calculus Level pending

\[\frac{\left (n! \right )^{4}}{\sqrt{n}(\sqrt{2\pi})^{3}\sqrt{\pi}\left (n^{n} \right )^{3}\left (e^{-n} \right )^{3}\left (\sqrt{\frac{n+\frac{1}{2}}{e}} \right )^{2n+1}\left (\sqrt{2n+\frac{1}{3}} \right )\sqrt{n+\frac{1}{6}+\frac{1}{72\left ( n+\frac{31}{90} \right )}-\frac{5929}{2332800\left ( n+\frac{3055123}{11205810} \right )^{3}}}}\]

Evaluate the expression above if \(n\) approaches infinity.

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