Product in terms of $n$ of $\frac{1}{2} \cdot \frac{3}{4} \cdot \frac{5}{6} \cdot \frac{7}{8} \cdot \cdots \cdot \frac{2n-1}{2n}$ What is the following product in terms of $n$?
$$\frac{1}{2} \cdot \frac{3}{4} \cdot \frac{5}{6} \cdot \frac{7}{8} \cdot \cdots \cdot \frac{2n-1}{2n}$$
Thank you.

As an alternative answer that has its uses, this product of fractions can be written as a definite integral:
$$\begin{align}
\frac12\frac34\frac56\frac78\dots\frac{2n-1}{2n}&=\frac{(2n-1)!!}{(2n)!!}\\
&=\frac{2}{\pi}\int_{0}^{\pi/2}\sin^{2n}{x}\,\mathrm{d}x
\end{align}$$
