I will present a simple proof of the Wallis formula using basic concepts of calculus. The Wallis formula states $\boxed{\lim_{n\to\infty} \frac{\big(n!\big)^22^{2n}}{(2n)!\sqrt{n}}=\sqrt{\pi}.\hspace{3pt}}\hspace{3pt}$ I will also show how the Wallis formula can be used to prove Stirling's formula, which says $\boxed{\lim_{n\to\infty} \frac{n!\hspace{1pt}e^{n}}{n^{n+1/2}\sqrt{2\pi}}=1.\hspace{3pt}}\hspace{3pt}$ Finally, I will talk about the Darboux formula, which states if $f$ is any Riemann integrable function in $[0,1]$ then $\boxed{\lim_{n\to\infty} \sum_{k=1}^{n} \frac{1}{n}f(k/n) = \int_{0}^{1} f(x) \hspace{3pt} dx. \hspace{3pt}}\hspace{3pt}$ Some examples on how these wonderful formulas can be used in practice will also be presented.
Thackeray 703
