Computing the integral
$$\int_0^{\infty}dz\prod_{k=1}^{\infty}\cos\frac{z}{2^k}$$
is fun once you realize $\prod_k\cos\frac{z}{2^k}=\frac{\sin z}z$. But, it's well-known that $\int_0^{\infty}\frac{\sin z}zdz=\frac{\pi}2$.

I've found the following variant intriguing and curious.

>**Question.** Is this valid? If not, what is the value of the integral?
$$\int_0^{\infty}dz\prod_{k=1}^{\infty}\cos\frac{z}{k}=\frac{\pi}4.$$