Euler's Reflection Formula

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The Formula

\[\Gamma(z)\Gamma(1-z) = \frac{\pi}{\sin \pi z}.\]

\[\large\sum_{x=1}^{1729} \left[\Gamma\left(\dfrac{1+2x}{2}\right)\Gamma\left(\dfrac{1-2x}{2}\right)\right]\]

Simplify the above summation and give your answer to correct four decimal places.

Clarification: \([\cdot]\) does not denote greatest integer function; they are just square brackets. Also, \(\Gamma(n)\) is the gamma function .

Bonus Question: Find the value of \[\displaystyle\sum_{x=1}^{1730} \left[\Gamma\left(\dfrac{1+2x}{2}\right)\Gamma\left(\dfrac{1-2x}{2}\right)\right].\]