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

,

or

.

(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.

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## About Simon

Hi. I'm Simon Rubinstein-Salzedo. I'm a mathematics postdoc at Dartmouth College. I'm also a musician; I play piano and cello, and I also sometimes compose music and study musicology. I also like to play chess and write calligraphy. This blog is a catalogue of some of my thoughts. I write them down so that I understand them better. But sometimes other people find them interesting as well, so I happily share them with my small corner of the world.