Basel Problem/Proof 10

Proof
From Mittag-Leffler Expansion for Hyperbolic Cotangent Function, we have:


 * $\displaystyle \frac 1 {2 z} \paren {\pi \map \coth {\pi z} - \frac 1 z} = \sum_{n \mathop = 1}^\infty \frac 1 {z^2 + n^2}$

We can write:

We therefore have:


 * $\displaystyle \sum_{n \mathop = 1}^\infty \frac 1 {n^2} = \lim_{z \mathop \to 0} \paren {\frac {\pi z \paren {e^{2 \pi z} + 1} - e^{2 \pi z} + 1} {2 z^2 \paren {e^{2 \pi z} - 1} } }$

We have at $z = 0$:

and:


 * $2 z^2 \paren {e^{2 \pi z} + 1} = 0$

So by L'Hopital's Rule:

giving:


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