User blog:Plain'N'Simple/A cool geometric connection regarding the fast-growing-hierarchy

Problem:

Suppose ABC is a right triangle, with BC=2 being the longest side and AC being twice the length of AB. Suppose further that we inscribe this triangle inside a circle.

What's the length of circle arc going from A to B?

Answer:

Arctan(1/2) = 1/f2(1)-1/f2(3)+1/f2(5)-1/f2(7)+1/f2(9)-1/f2(11)...

(where f2(n)=n*2n is the second function of the FGH).

Cool, eh? This kind of crazy connections is exactly the reason I love mathematics so much.

Note:

I discovered this connection when doing a physics problem that involved the arctangent of 1/2. I didn't have a calculator near me and was too lazy to get up and fetch one, so I decided to calculate the sum for arctan 1/2 on paper. The numbers seemed familiar (1/8, 1/24, 1/160, 1/896...) and then it clicked.