# Basel Problem/Proof 7

## Theorem

$\displaystyle \map \zeta 2 = \sum_{n \mathop = 1}^\infty {\frac 1 {n^2} } = \frac {\pi^2} 6$

where $\zeta$ denotes the Riemann zeta function.

## Proof

By Fourier Series of $x^2$, for $x \in \openint {-\pi} \pi$:

$\displaystyle x^2 = \frac {\pi^2} 3 + \sum_{n \mathop = 1}^\infty \paren {\paren {-1}^n \frac 4 {n^2} \cos n x}$

Letting $x \to \pi$ from the left:

 $\displaystyle \pi^2$ $=$ $\displaystyle \frac {\pi^2} 3 + \sum_{n \mathop = 1}^\infty \paren {\paren {-1}^n \frac 4 {n^2} \cos \pi x}$ $\displaystyle \pi^2$ $=$ $\displaystyle \frac {\pi^2} 3 + \sum_{n \mathop = 1}^\infty \paren {\paren {-1}^n \paren {-1}^n \frac 4 {n^2} }$ Cosine of Multiple of $\pi$ $\displaystyle \pi^2$ $=$ $\displaystyle \frac {\pi^2} 3 + 4 \sum_{n \mathop = 1}^\infty \frac 1 {n^2}$ $\displaystyle \leadsto \ \$ $\displaystyle \frac {2 \pi^2} 3$ $=$ $\displaystyle 4 \sum_{n \mathop = 1}^\infty \frac 1 {n^2}$ $\displaystyle \leadsto \ \$ $\displaystyle \sum_{n \mathop = 1}^\infty \frac 1 {n^2}$ $=$ $\displaystyle \frac {\pi^2} 6$

$\blacksquare$