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Calculus Level 3

n=1(4n2n)1=X+πY2ZZln(1+Z2)\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 XX, YY, and ZZ. Find XYZ\sqrt{XYZ}.

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

n=1(4n2n)1=1(42)+1(84)+1(126)+=16+170+1924+.\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}

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