Worked solution: Euler's proof via the sine product formula (1734–1735)

Step 1 of 6: The target sum and Euler's idea

In plain words

Imagine adding $1$ plus $\frac{1}{4}$ plus $\frac{1}{9}$ plus $\frac{1}{16}$ and so on forever: the terms shrink fast, so the total settles on one fixed number, yet nobody could name that number for almost a century. Euler's trick was to sneak up on the very same number from a completely different direction — through the trigonometric function $\sin x$ — and read off its value by comparing two different descriptions of the same object.

\zeta(2)=\sum_{n=1}^{\infty}\frac{1}{n^2}=1+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+\cdots

Detailed analysis

The Basel problem asks for the exact value of this sum, known numerically since the 1600s but with no closed form. Euler's idea was to find a second formula for the same number from the Maclaurin series of $\sin x$ , then match it to a completely different-looking product built from the zeros of $\sin x$ .

Terms in this step

$\zeta(2)$: Standard shorthand for this particular infinite sum: the value at $2$ of the Riemann zeta function $\zeta(s)=\sum_{n=1}^\infty 1/n^s$ .
Closed form: An expression built from a finite number of familiar constants and operations, such as $\pi^2/6$ , as opposed to an infinite, unevaluated process.

Knowledge used in this step

Numerical series and convergence tests