Are these new series formulae for $\zeta(2)$?

sequences-and-series riemann-zeta

Please see the following answer for the proof of the fact

$\displaystyle \sum_{n=1}^\infty \frac{x^n}{n^2 \binom{2n}{n}}=2\left[\sin^{-1} \frac{\sqrt{x}}{2} \right]^2$

Just note that in the result given in answer, you need to replace $x$ by $\frac{\sqrt{x}}{2}$ and also make sure that $0 \leq x \leq 4$

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