<s>[INST] $\{f(n) = \frac{1}{4n \tan \frac{\pi}{n}} \}$ is a monotonically increasing sequence Could anyone is able to give me a hint why $\{f(n) = \frac{1}{4n \tan \frac{\pi}{n}} \}$ is a monotonically increasing sequence?
 [/INST] Function
$$f(x)=\frac{1}{4x\tan{\frac{\pi}{x}}}$$ is increasing, since $f'(x)>0$, which is easily proved:
$$f'(x)=\frac{-4\tan{\frac{\pi}{x}}+\frac{4\pi}{x\cos^2{\frac{\pi}{x}}}}{4x^2\tan^2{\frac{\pi}{x}}}>0$$ 
is equivalent with $$-4\tan{\frac{\pi}{x}}+\frac{4\pi}{x\cos^2{\frac{\pi}{x}}}>0,$$
which is again equivalent with inequality
$$-4\sin{\frac{\pi}{x}}\cos{\frac{\pi}{x}}+4\frac{\pi}{x}>0,$$
that holds, since $\sin{t}<t$ and $\cos{t}<1$ for every $t\in\mathbb{R}$.
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