$\int_0^{\infty} \frac{\sin x}{x}\frac{\sin \frac{x}{3}}{\frac{x}{3}} dx = \frac{\pi}{2}$ and more general results? Problem: evaluate that $\int_0^{\infty} \frac{\sin x}{x}\frac{\sin \frac{x}{3}}{\frac{x}{3}} dx = \frac{\pi}{2}$ and prove if a more general case, $\int_0^{\infty} \frac{\sin x}{x}\frac{\sin \frac{x}{3}}{\frac{x}{3}} \frac{\sin \frac{x}{5}}{\frac{x}{5}} ...  \frac{\sin \frac{x}{2n+1}}{\frac{x}{2n+1}} dx = \frac{\pi}{2}$ holds?
So it's a rather well known results to prove Evaluating the integral $\int_0^\infty \frac{\sin x} x \,\mathrm dx = \frac \pi 2$?
and I use a computer program to validate that the general case does appear to hold, but not sure how to prove it..

These are Borwein Integrals and the behavior of exactly equaling $\pi/2$ stops at $2n+1 = 15$.
