Even And Odd Partition?

Let $x = \lfloor \pi P \rfloor$ , then we have:

$\begin{aligned} P & = \frac{2}{1} \cdot{} \frac{2}{3} \cdot{} \frac{4}{3} \cdot{} \frac{4}{5} \cdot{} \frac{6}{5} \cdot{} \frac{6}{7} \cdot{} \frac{8}{7} \cdot{} \frac{8}{9} ... \cdot{} \infty \\ & = \lim_{m \to \infty} \prod_{k=1}^m \frac{2k}{2k-1} \cdot{} \frac{2k}{2k+1} \\ & = \lim_{m \to \infty} \frac{2^m m! \cdot{} 2^m m!}{(2m)!} \cdot{} \frac{2^m m! \cdot{} 2^m m!}{(2m+1)!} \\ & = \lim_{m \to \infty} \frac{2^{4m} (m!)^4}{(2m)!(2m+1)!} \quad \quad \small \color{#3D99F6}{\text{Using Stirling's approximation } n! \approx \sqrt{2\pi} e^{-n} n^{n+\frac{1}{2}}} \\ & = \lim_{m \to \infty} \frac{2^{4m} \left(\sqrt{2\pi} e^{-m} m^{m+\frac{1}{2}}\right)^4}{\sqrt{2\pi} e^{-2m} (2m)^{2m+\frac{1}{2}} \sqrt{2\pi} e^{-2m-1} (2m+1)^{2m+\frac{3}{2}}} \\ & = \lim_{m \to \infty} \frac{2^{2m+\frac{1}{2}} m^{2m+\frac{3}{2}} \cdot{} \pi e}{(2m+1)^{2m+\frac{3}{2}}} \\ & = \lim_{m \to \infty} \frac{(2m)^{2m+\frac{3}{2}} \cdot{} \pi e}{2(2m+1)^{2m+\frac{3}{2}}} \\ & = \lim_{m \to \infty} \frac{\pi e}{2\left(1+\frac{1}{2m} \right)^{2m+\frac{3}{2}}} \\ & = \frac{\pi e}{2 e} = \frac{\pi}{2} \end{aligned}$

$\Rightarrow x = \lfloor \pi P \rfloor = \left \lfloor \pi \cdot{} \dfrac{\pi}{2} \right \rfloor = \lfloor 4.9348 \rfloor = \boxed{4}$