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For $I_n = \int_0^{\pi/2} \sin^n\theta \, d\theta$ it is possible to show (using integration by parts and $\sin^2\theta = 1-\cos^2\theta$) that: $nI_n = (n-1)I_{n-2}$

We can then repeatedly apply this relation to $\frac{I_{2n+1}}{I_{2n}}$ to show: $$\frac{I_{2n+1}}{I_{2n}} = \frac{2^{4n+1}(n!)^4}{\pi (2n)!(2n+1)!}$$

This ratio converges to $1$ since $I_{2n+2}/I_{2n} \rightarrow 1$ (easy to show with the above recursive relation) which is always larger than $I_{2n+1}/I_{2n}$. Since $I_{2n}/I_{2n} = 1$, we can sandwich the desired ratio.

I also know that (from Show that $n!e^n/n^{n+1/2} \leq e^{1/(4n)}C$) that: $r_n = n!e^n/n^{n+1/2} \leq e^{1/(4n)}C$ for all $n$ and for $C = \lim_{n\rightarrow \infty} n!e^n/n^{n+1/2}$. Now I want to show Stirling's formula:

$$n! \sim \sqrt{2\pi}n^{n+1/2}/e^n$$

What I've tried

Using the upper bound on $r_n$ it is clear that all that remains is to show that $C = \sqrt{2\pi}$. I have seen trick on related problems involving expressing the integral $I_n$ in terms of the beta function, but I think another trick is needed as well (perhaps one involving introducing a term $e^{i\theta}$ but I am struggling to get anything more specific than that.

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Note that you know a little bit more from the linked question - that $n!/(n^{n+1/2} e^{-n})$ decreases to $C$.

Using this and the upper bound you have, note that $$ \frac{I_{2n+1}}{I_{2n}} \le \frac{2^{4n + 1} C^4 n^{4n + 2} e^{-4n} e^{1/n}}{\pi \cdot C(2n)^{2n+1/2}e^{-2n} \cdot C(2n+1)^{2n + 3/2} e^{-2n - 1}} \\ = \frac{C^2}{2\pi} \frac{e^{1+1/n}}{(1 + 1/2n)^{2n + 3/2}} =: a_n$$

Since $I_{2n+1}/I_{2n} \to 1,$ we can conclude that $\liminf a_n \ge 1$. But $\liminf a_n = \frac{C^2}{2\pi}$, thus telling us that $C \ge \sqrt{2\pi}$.

Similarly, we can develop the lower bound $$ \frac{I_{2n+1}}{I_{2n}} \ge \frac{C^2}{2\pi} \frac{e}{e^{1/8n + 1/8n + 4} (1+1/2n)^{2n + 3/2}} =: b_n, $$

and argue that $\frac{C^2}{2\pi} = \limsup b_n \le 1,$ ergo $C \le \sqrt{2\pi}$.

Thus we have shown that $\sqrt{2\pi} \le C \le \sqrt{2\pi},$ which of course implies that $C = \sqrt{2\pi}$.

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I believe this is exactly what you are looking for: https://en.wikipedia.org/wiki/Wallis%27_integrals#Deducing_Stirling's_formula.

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