Wallis's Product

Theorem

 * $ \displaystyle \prod_{n=1}^{\infty} \frac{2n}{2n-1} \cdot \frac{2n}{2n+1} = \frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5} \cdot \frac{6}{5} \cdot \frac{6}{7} \cdot \frac{8}{7} \cdot \frac{8}{9} \cdots = \frac{\pi}{2}$

Wallis's Original Proof
Wallis, of course, had no recourse to Euler's techniques.

He did this job by comparing $\displaystyle \int_0^\pi \sin^n x dx$ for even and odd values of $n$, and noting that for large $n$, increasing $n$ by 1 makes little change.