When I first saw:
$$\frac{\pi}{4}=1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}\mp\cdots.$$
I was puzzled that such an expression could have anything to do with circles. There are tons more and it seems like a very large amount of summations will end up having to do with $\pi$. My question is how come $\pi$ finds its way into so many formulas, and could the above equation be thought of geometrically.