Basel Problem/Proof 4

Theorem

 * $\displaystyle \zeta \left({2}\right) = \sum_{n \mathop = 1}^\infty \frac 1 {n^2} = \frac {\pi^2} 6$

where $\zeta$ denotes the Riemann zeta function.

Proof
Let $n$ be a positive integer.

We have:

We also have:

So we can deduce:


 * $\displaystyle \sum_{n \mathop = 0}^\infty \frac 1 {\left({2n + 1}\right)^2} = \frac {\pi^2} 8$

Finally note that: