The Basel Problem

The Basel problem asks for the exact sum of the reciprocals of the squares.

$$\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}$$

Why It Is Surprising

The left side is a sum of rational numbers. The right side involves $\pi$, a geometric constant. There is no obvious reason they should be equal.

Euler solved this in 1734 at age 28 by noticing a connection between the sine function and infinite products. It made him famous.