Wallis's Product

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

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

Also presented as
This result can also be seen presented as:
 * $\ds \prod_{n \mathop = 1}^\infty \frac n {n - \frac 1 2} \cdot \frac n {n + \frac 1 2} = \frac \pi 2$