$\sum_{n=1}^\infty \binom{4n}{2n}^{-1}=X+\frac{\pi}{\sqrt{Y}}-\frac{2}{Z\sqrt{Z}}\ln\left(\frac{1+\sqrt{Z}}{2}\right)$

The equation above holds true for rational numbers $X$ , $Y$ , and $Z$ . Find $\sqrt{XYZ}$ .

**
Note
**
:
$\binom{\cdot}{\cdot}$
is the
binomial coefficient
. The first few terms of the series are as follows:

$\begin{aligned} \sum_{n=1}^\infty \binom{4n}{2n}^{-1} &= \frac{1}{\binom{4}{2}} + \frac{1}{\binom{8}{4}} + \frac{1}{\binom{12}{6}} + \cdots \\ &= \frac{1}{6}+\frac{1}{70}+\frac{1}{924}+\cdots. \end{aligned}$

The answer is 9.

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The solution could be found here : see Theorem 3.8 on page 11. The last thing to be considered here is to ensure that the lower summation limits are not identical: 1 - here and 0 - via the link formulae.