Wallis's Product

Theorem

 * $ \displaystyle \prod_{n \mathop = 1}^\infty \frac {2 n} {2 n - 1} \cdot \frac {2 n} {2 n + 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 techniques.

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