What does $\left[\pi \cdot \frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5} \cdot \frac{6}{5} \ldots \ldots \infty\right]$ equals?

Question

I recently came across this interesting question

Find the value of $\left[\pi(\frac{2}{1}) (\frac{2}{3} )(\frac{4}{3}) (\frac{4}{5}) (\frac{6}{5})(\frac{6}{7})(\frac{8}{7})(\frac{8}{9}) \cdots (\infty)\right]$

My Approach: I could see a pattern forming like $$\left[2\pi\ \cdot \left(1 - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{3}\right) \cdot \left(1 - \frac{1}{5}\right) \cdot \left(1 + \frac{1}{5}\right) \ldots \ldots \infty \right]$$

But not able to see how to proceed further