Pi as an infinite product - The Wallis product

The number π has many surprising and interesting properties. As it is well-known, it is defined as the ratio between the perimeter and the diameter of a circle. There are many interesting identities of this number, that can be expressed as an infinite sum or product of rational numbers, some of these expressions are very beautiful.

Image from pixbay

A few days ago I wrote a post about the series



The Wallis' product, gives us an expression of π as an infinity product:



In the following lines I will show a proof of this identity. It requires knowledge of basic calculus and it is a nice exercise.

If we compute the integral of sinⁿ(x) by parts we have:

for the details of this computation, you can see the recent post done by @masterwu.

If we take limits of the integrals between 0 and π/2 of the previous expression we get the identity:

We consider odd and even powers of sin x separately. After iterating the computations, we get, for the odd case:

Since we get the identity:

In the even case, we have the expression



Combining the last two identities we have the equality:



We would like to take the limit when n goes to infinity in this last expression. In order to do that, we have to consider that 0≤ sin x≤1, when 0≤ x ≤ π/2. So



Hence, it follows:



From identity (1) follows:



Therefore:



After all this analysis we can conclude



So, after taking the limit when n goes to infinity in the expression (2), we get the desired identity:

References:
Edwards, C.H., "Advanced calculus of several variables", Academic Press, 1973.
https://en.wikipedia.org/wiki/Wallis_product

The source of the first image is Pixbay and all the formulae of this post were typed by myself in LaTeX.

My latest post about mathematics are:
The look and say sequence,
The Banach-Tarski paradox,
Euler's solution of the Basel problem.