Towards an Expression for pi II

Continuing from this post

We start with $latex \cos( \dfrac{\xi}{2})\cos(\dfrac{\xi}{4}) … \cos(\dfrac{\xi}{2^n})$. Recall the double angle identity for sin: $latex \sin 2 \theta = 2\sin \theta \cos \theta $. We will use this a lot.

Multiply our expression by $latex \sin(\dfrac{\xi}{2^n})$. Then we have

$latex \cos( \dfrac{\xi}{2})\cos(\dfrac{\xi}{4}) … \cos(\dfrac{\xi}{2^n})\sin(\dfrac{\xi}{2^n})$

Using the double angle identity, we can reduce this:

$latex = \dfrac{1}{2} \cos( \dfrac{\xi}{2})\cos(\dfrac{\xi}{4}) … \cos(\dfrac{\xi}{2^{n-1}})sin(\dfrac{\xi}{2^{n-1}}) =$
$latex = \dfrac{1}{4} \cos( \dfrac{\xi}{2})\cos(\dfrac{\xi}{4}) … \cos(\dfrac{\xi}{2^{n-2}})\sin(\dfrac{\xi}{2^{n-2}}) =$
$latex …$
$latex = \dfrac{1}{2^{n-1}}\cos(\xi / 2)\sin(\xi / 2) = \dfrac{1}{2^n}\sin(\xi)$

So we can rewrite this as

$latex \cos( \dfrac{\xi}{2})\cos(\dfrac{\xi}{4}) … \cos(\dfrac{\xi}{2^n}) = \dfrac{\sin \xi}{2^n \sin( \dfrac{\xi}{2^n} )}$ for $latex \xi \not = k \pi$

Because we know that $latex lim_{x \to \infty} \dfrac{\sin x}{x} = 1$, we see that $lim_{n \to \infty} \dfrac{\xi / 2^n}{\sin(\xi / 2^n)} = 1$. So we see that

$latex \cos( \dfrac{\xi}{2})\cos(\dfrac{\xi}{4}) … = \dfrac{\xi}{\xi}$
$latex \xi = \dfrac{\sin(\xi)}{\cos(\dfrac{\xi}{2})\cos(\dfrac{\xi}{4})…}$

Now we set $latex \xi := \pi /2$. Also recalling that $latex \cos(\xi / 2 ) = \sqrt{ 1/2 + 1/2 \cos \xi}$. What do we get?

$latex \dfrac{\pi}{2} = \dfrac{1}{\sqrt{1/2} \sqrt{ 1/2 + 1/2 \sqrt{1/2} } \sqrt{1/2 + 1/2 \sqrt{ 1/2 + 1/2 \sqrt{1/2} …}}}$

This is pretty cool. It’s called Vieta’s Formula for $latex \dfrac{\pi}{2}$. It’s also one of the oldest infinite products.

This entry was posted in Expository, Math.REC, Mathematics and tagged , , , . Bookmark the permalink.

Leave a Reply