Note: this proof has a fatal error.
There’s a reasonably well-known proof that , which can be written in one line:
But I’ve always found this one unsatisfying because what does that integral have to do with anyway? As it turns out, enters through the integral . But let’s say I’m a purist and think that is about circles. Can I do better? (Of course I can. If I couldn’t I wouldn’t be writing this post.) Start by observing that , which can be shown by explicit computation: and .
Edited to add, November 30: of course I just showed here that . This is what happens when you do arithmetic in your head…
Now, the sine function is given by the alternating series and in particular by the alternating series test. Applying this with gives .
Taking the inverse sine of both sides, .
Finally, we have . This is a geometric fact that goes back to Euclid’s construction of the hexagon. So .
On related notes:
- Noam Elkies, Why is π2 so close to 10?
- Alejandro Morales, Igor Pak, and Greta Panova, Why is π < 2φ? (This is not a particularly good approximation, but it actually admits a combinatorial proof in terms of Fibonacci and Euler numbers.)