Formula for the ratio $\frac{\Gamma\left(n + \frac{1}{2}\right)}{\Gamma(n + 1)}$ of two values of the Gamma function

Question

Show that $$\frac{\Gamma\left(n + \frac{1}{2}\right)}{\Gamma(n + 1)} = \frac{1 \cdot 3 \cdot \cdots (2 n - 3) (2 n - 1)}{2 \cdot 4 \cdot \cdots (2 n - 2) \cdot 2n} .$$

I have proved that $$\Gamma\left(n + \frac{1}{2}\right) = \frac{1 \cdot 3 \cdot \cdots (2 n - 3) (2 n - 1) \sqrt{\pi}}{2^n} \qquad \textrm{and} \qquad\Gamma(n+1) = n! ,$$ but when I divide, I am not able to proceed. Could you please help me?

Answer 1

You may

1. Define $f(n)$ as $\frac{\Gamma(n+1/2)}{\Gamma(n+1)}$ and $g(n)$ as $\frac{(2n-1)!!}{(2n)!!}$
2. Compute $\frac{f(n+1)}{f(n)}$ (via $\Gamma(z+1)=z\,\Gamma(z)$) and $\frac{g(n+1)}{g(n)}$, then check they are equal
3. Compare $f(1)$ with $g(1)$ to derive that $f(n)$ is a constant multiple of $g(n)$ for any $n\in\mathbb{N}^+$.