What sequence is this $\prod_{i=1}^n \frac{2i}{2i + 1} = $? I'm trying to find a closed form solution for $\prod_{i=1}^n \frac{2i}{2i + 1} $, but I'm confused what kind of sequence this is. Does it have a closed form?

After @Lemoine's answer, you can also write
$$P_n=\prod_{i=1}^n \frac{2i}{2i + 1}=\frac{\sqrt{\pi }\,\, \Gamma (n+1)}{2\,\, \Gamma \left(n+\frac{3}{2}\right)}$$ Take logarithms, use Stirling approximation twice and continue with Taylor series to obtain
$$\log(P_n)=-\frac{1}{2} \log \left(\frac{4 n}{\pi }\right)-\frac{3}{8 n}+\frac{1}{8 n^2}+O\left(\frac{1}{n^3}\right) $$
$$P_n=e^{\log(P_n)}=\frac{\sqrt{\pi }}{2 \sqrt{n}}\,\,\exp\Big[-\frac{3}{8 n}+\frac{1}{8 n^2}+O\left(\frac{1}{n^3}\right) \Big]$$
Try it for $n=10$; the exact value is
$$P_{10}=\frac{262144}{969969}\sim 0.270260$$ while the above truncated expansion gives $0.270272$.
