I had seen the Wallis product, but I did not know a derivation of it until a few weeks ago, when I discovered this gem when reading the wonderful book Sources in the Development of Mathematics by Ranjan Roy. I’m storing this example to use next time I teach calculus, as it requires nothing more than integration by parts.
Like in the case of many other such things in mathematics, I don’t know what the Wallis product actually is; I can only discover it by going through a derivation. So, I won’t give away the answer until we get there. But the goal is to write down an infinite product for .
In order to do this, we evaluate the integral . The first few are easy:
I can keep working out more of these, but now it’s time to tackle the general case, using integration by parts. We assume , and we compute , where and , so that
Solving for , we get . Now it’s easy to compute more of the ‘s:
and in general
Now, note that for any , , since whenever . Furthermore, since , this means that eventually and get very close together. So
(If we wanted, we could be more precise, and give two inequalities for , but it doesn’t add much value, in my opinion, to do that here.)
Taking the limit, we get
This is the Wallis product.