Wallis's Product/Original Proof

Theorem

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

Proof
From the Reduction Formula for Integral Power of Sine Function, we have:
 * $\displaystyle (1): \qquad \int \sin^n x \mathrm \,\mathrm d x = - \frac 1 n \sin ^{n-1} \cos x + \frac {n-1} n \int \sin^{n-2} x\, \mathrm d x$

Let $I_n$ be defined as:
 * $\displaystyle I_n = \int_0^{\pi / 2} \sin^n x \,\mathrm d x$

As $\displaystyle \cos \frac \pi 2 = 0$ from Shape of Cosine Function, we have from $(1)$ that:
 * $\displaystyle (2): \qquad I_n = \frac {n-1} n I_{n-2}$

To start the ball rolling, we note that:
 * $\displaystyle I_0 = \int_0^{\pi / 2} \,\mathrm d x = \frac \pi 2 \qquad \qquad I_1 = \int_0^{\pi / 2} \sin x \, \mathrm d x = \left[{- \cos x}\right]_0^{\pi / 2} = 1$

We need to separate the cases where the subscripts are even and odd:

By Shape of Sine Function, we have that on $0 \le x \le \dfrac \pi 2$ we have that $0 \le \sin x \le 1$.

Therefore:
 * $\displaystyle 0 \le \sin^{2n+2} x \le \sin^{2n+1} x \le \sin^{2n} x$

It follows from Relative Sizes of Definite Integrals that:
 * $\displaystyle 0 < \int_0^{\pi/2} \sin^{2n+2} x \ \mathrm d x \le \int_0^{\pi/2} \sin^{2n+1} x \ \mathrm d x \le \int_0^{\pi/2} \sin^{2n} x \ \mathrm d x$

That is:
 * $\displaystyle (3) \qquad 0 < I_{2n+2} \le I_{2n+1} \le I_{2n}$

By $(2)$ we have:
 * $\displaystyle \frac {I_{2n_2}}{I_{2n}} = \frac {2n+1}{2n+2}$

Dividing $(3)$ through by $I_{2n}$ then, we have:
 * $\displaystyle \frac {2n+1}{2n+2} \le \frac {I_{2n+1}}{I_{2n}} \le 1$

It follows then that:
 * $\displaystyle \frac {I_{2n+1}}{I_{2n}} \to 1$ as $n \to \infty$

which is equivalent to:
 * $\displaystyle \frac {I_{2n}}{I_{2n+1}} \to 1$ as $n \to \infty$

Now we take $(B)$ and divide it by $(A)$ to get:


 * $\displaystyle \frac {I_{2n+1}}{I_{2n}} = \frac 2 1 \cdot \frac 2 3 \cdot \frac 4 3 \cdot \frac 4 5 \cdots \frac {2n} {2n-1} \cdot \frac {2n} {2n+1} \cdot \frac 2 \pi$

So:
 * $\displaystyle \frac \pi 2 = \frac 2 1 \cdot \frac 2 3 \cdot \frac 4 3 \cdot \frac 4 5 \cdots \frac {2n} {2n-1} \cdot \frac {2n} {2n+1} \cdot \left({\frac {I_{2n}}{I_{2n+1}}}\right)$

Taking the limit as $n \to \infty$ gives the result.