Factorial and product series

Question

Help me prove this equation is right

$$n!=\frac{\prod_{k=0}^{n-1}\left(-\frac{1}{2}-k\right)}{\prod_{k=1}^{n}\left(\frac{1}{2k}-1\right)}$$

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Answer 1

Rewrite as

$$\frac{\prod_{k=0}^{n-1}\left(-\frac{1}{2}-k\right)}{\prod_{k=1}^{n}\left(\frac{1}{2k}-1\right)}$$

$$=\prod_{k=1}^n\frac{\left(-\frac{1}{2}-(k-1)\right)}{\left(\frac{1}{2k}-1\right)}$$

This simplifies to

$$=\prod_{k=1}^n\frac{\left(\frac{1-2k}{2}\right)}{\left(\frac{1-2k}{2k}\right)}$$

$$=\prod_{k=1}^n\ k$$