Wallis's Product/Original Proof

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Theorem

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


Proof

From the Reduction Formula for Integral of Power of Sine, we have:

$\ds (1): \quad \int \sin^n x \rd x = - \frac 1 n \sin^{n - 1} x \cos x + \frac {n - 1} n \int \sin^{n - 2} x \rd x$


Let $I_n$ be defined as:

$\ds I_n = \int_0^{\pi / 2} \sin^n x \rd x$

As $\cos \dfrac \pi 2 = 0$ from Shape of Cosine Function, we have from $(1)$ that:

$(2): \quad I_n = \dfrac {n-1} n I_{n - 2}$


To start the ball rolling, we note that:

$\ds I_0 = \int_0^{\pi / 2} \rd x = \frac \pi 2$
$\ds I_1 = \int_0^{\pi / 2} \sin x \rd x = \bigintlimits {-\cos x} 0 {\pi / 2} = 1$

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

\(\ds I_{2 n}\) \(=\) \(\ds \frac {2 n - 1} {2 n} I_{2 n - 2}\)
\(\ds \) \(=\) \(\ds \frac {2 n - 1} {2 n} \cdot \frac {2 n - 3} {2 n - 2} I_{2 n - 4}\)
\(\ds \) \(=\) \(\ds \cdots\)
\(\ds \) \(=\) \(\ds \frac {2 n - 1} {2 n} \cdot \frac {2 n - 3} {2 n - 2} \cdot \frac {2 n - 5} {2 n - 4} \cdots \frac 3 4 \cdot \frac 1 2 I_0\)
\(\text {(A)}: \quad\) \(\ds \) \(=\) \(\ds \frac {2 n - 1} {2 n} \cdot \frac {2 n - 3} {2 n - 2} \cdot \frac {2 n - 5} {2 n - 4} \cdots \frac 3 4 \cdot \frac 1 2 \cdot \frac \pi 2\)


\(\ds I_{2 n+1}\) \(=\) \(\ds \frac {2 n} {2 n + 1} I_{2 n - 1}\)
\(\ds \) \(=\) \(\ds \frac {2 n} {2 n + 1} \cdot \frac {2 n - 2} {2 n - 1} I_{2 n - 3}\)
\(\ds \) \(=\) \(\ds \cdots\)
\(\ds \) \(=\) \(\ds \frac {2 n} {2 n + 1} \cdot \frac {2 n - 2} {2 n - 1} \cdot \frac {2 n - 4} {2 n - 3} \cdots \frac 4 5 \cdot \frac 2 3 I_1\)
\(\text {(B)}: \quad\) \(\ds \) \(=\) \(\ds \frac {2 n} {2 n + 1} \cdot \frac {2 n - 2} {2 n - 1} \cdot \frac {2 n - 4} {2 n - 3} \cdots \frac 4 5 \cdot \frac 2 3\)


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

$0 \le \sin x \le 1$

Therefore:

$0 \le \sin^{2 n + 2} x \le \sin^{2 n +1} x \le \sin^{2 n} x$

It follows from Relative Sizes of Definite Integrals that:

$\ds 0 < \int_0^{\pi / 2} \sin^{2 n + 2} x \rd x \le \int_0^{\pi / 2} \sin^{2 n + 1} x \rd x \le \int_0^{\pi / 2} \sin^{2 n} x \rd x$

That is:

$(3): \quad 0 < I_{2 n + 2} \le I_{2 n + 1} \le I_{2 n}$


By $(2)$ we have:

$\dfrac {I_{2 n + 2} } {I_{2 n} } = \dfrac {2 n + 1} {2 n + 2}$

Dividing $(3)$ through by $I_{2n}$ then, we have:

$\dfrac {2 n + 1} {2 n + 2} \le \dfrac {I_{2 n + 1}} {I_{2 n}} \le 1$

By Squeeze Theorem, it follows that:

$\dfrac {I_{2 n + 1} } {I_{2 n} } \to 1$ as $n \to \infty$

which is equivalent to:

$\dfrac {I_{2 n} } {I_{2 n + 1} } \to 1$ as $n \to \infty$

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

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

So:

$\dfrac \pi 2 = \dfrac 2 1 \cdot \dfrac 2 3 \cdot \dfrac 4 3 \cdot \dfrac 4 5 \cdots \dfrac {2 n} {2 n - 1} \cdot \dfrac {2 n} {2 n + 1} \cdot \paren {\dfrac {I_{2 n} } {I_{2 n + 1} } }$

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

$\blacksquare$


Source of Name

This entry was named for John Wallis.


Sources