Supposedly the answer is 1 but I have no idea how to evaluate this thing analytically.
$$f(n) = \frac{2}{\pi} \int_{0}^{\infty} 2\cos(x) \cdot \frac{\sin(x)}{x} \cdot \frac{\sin(x/3)}{x/3} \cdot \cdots \cdot \frac{\sin(x/(2n+1))}{x/(2n+1)} dx$$
Any help would be appreciated.
The conjecture $f(n)=1$ is only correct for $n\leq 55$, see H. Schmid, Two curious integrals and a graphic proof. For $n=56$ an analytical calculation using the Poisson summation formula gives $$f(56)= 1 − 1.484870809 \cdot 10^{−138},$$ see More remarkable sinc integrals and sums.
Schmid remarks:
When this [deviation from unity] was recently verified by a researcher using a computer algebra package, he concluded that there must be a “bug” in the software. It is not a bug, though; this series of integrals really only results in 1 up to a certain point, and then breaks down. This astonishes most mathematically educated readers...
