Central Binomial representation for zeta(2)

Central Binomial representation for zeta(2)

In this post we will prove the following nice series representation for $\zeta(2)$ :

$\begin{align*} \zeta(2)=\frac53 \sum_{n=0}^\infty \frac{(-1)^n\binom{2n}{n}}{2^{4n}(2n+1)^2} \end{align*}$

Click here for the proof.

We used the previous established result

$\begin{align*} \operatorname{Li}_2\left(\phi^{-2})\right)=\frac{\pi^2}{15}-\ln^2(\phi) \end{align*}$

Click here for the proof.

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