Found an elementary proof of Binet's Formula for the Gamma Function s below:

The present note presents an elementary proof of the following important result of J. P. M. Binet [3, p. 249].

Theorem 1. For $x>0$ we have

$$

\Gamma(x+1)=\left(\frac{x}{\mathrm{e}}\right)^{x} \sqrt{2 \pi x} \cdot \mathrm{e}^{\theta(x)}

$$

where

$$

\theta(x)=\int_{0}^{\infty}\left(\frac{1}{\mathrm{e}^{t}-1}-\frac{1}{t}+\frac{1}{2}\right) \mathrm{e}^{-x t} \frac{1}{t} d t

$$

Here $\Gamma$ denotes the gamma function defined by

$$

\Gamma(x)=\int_{0}^{\infty} t^{x-1} \mathrm{e}^{-t} d t

$$

Since $\lim _{x \rightarrow \infty} \theta(x)=0$, from $(1)$ we immediately obtain Stirling's formula

$$

n !=\Gamma(n+1) \sim\left(\frac{n}{\mathrm{c}}\right)^{n} \sqrt{2 \pi n} .

$$

Binet's formula can also be used to prove a more precise version of Stirling's asymptotic expansion

$$\log \frac{n !}{(n / \mathrm{e})^{n} \sqrt{2 \pi n}}=\sum_{j=1}^{\infty} \frac{B_{2 j}}{2 j(2 j-1) n^{2 j-1}}=\frac{1}{12 n}-\frac{1}{360 n^{3}}+\frac{1}{1260 n^{5}}-\cdots$$

where the $B_{2 j}$ 's denote the Bernoulli numbers defined by

$$

\frac{1}{\mathrm{e}^{t}-1}-\frac{1}{t}+\frac{1}{2}=\sum_{j=1}^{\infty} \frac{B_{2 j}}{(2 j) !} t^{2 j-1}

$$

For, by problem 154 in Part I, Chapter 4 of [2], the inequalities

$$

\sum_{j=1}^{2 N} \frac{B_{2 j}}{(2 j) !} t^{2 j-1}<\frac{1}{\mathrm{e}^{t}-1}-\frac{1}{t}+\frac{1}{2}<\sum_{j=1}^{2 N+1} \frac{B_{2 j}}{(2 j) !} t^{2 j-1}

$$

Sasvari, Z. (1999). An Elementary Proof of Binet's Formula for the Gamma Function. *The American Mathematical Monthly,* *106*(2), 156-158. doi:10.2307/2589052