I am attempting to show that there does not exist an N past which every open unit interval (k, k+1) -where k is an integer- contains a zero of the following function:

$h(x)=\sum_{n=2}^{[\sqrt(x)]} \frac{\cot(x\pi/n)}{n}+\frac{\cot((x+2)\pi/n)}{n}$

Where the $[\sqrt(x)]$ is the lowest integer of the square root of $x$. Any thoughts? I had figured that I could consider the interval $(i^2, (i+1)^2)$ in which h(x) is described by the function

$h(x)=h_n(x)=\sum_{n=2}^{i} \frac{\cot(x\pi/n)}{n}+\frac{\cot((x+2)\pi/n)}{n}$

Then try to see on what unit intervals (k, k+1) in here contain a zero of $h_n$ then try to show that $((i+1)^2, (i+2)^2)$ also has such an interval, but I am running out of ideas.