Landau notation on $f(n) = \frac{1}{4n \tan(\frac{\pi}{n})}$ Does someone would have time to show me how to use the Landau notation "Big O"? A useful example could be on $f(n) = \frac{1}{4n \tan(\frac{\pi}{n})}$.

One has, by the Taylor series expansion, as $x \to 0$,
$$
\frac{1}{1+x}=1+O(x),\qquad \tan x=x+O(x^3),\tag1
$$ then
$$
\frac{x}{\tan x}=\frac{x}{x+O(x^3)}=\frac1{1+O(x^2)}=1+O(x^2)\tag2
$$ Hence, as $n \to \infty$, we have $\dfrac{\pi}n \to 0$, and
$$
\begin{align}
f(n) := \frac{1}{4n \tan(\frac{\pi}{n})}
=\frac1{4\pi}\cdot \frac{\frac{\pi}{n}}{\tan(\frac{\pi}{n})}
=\frac1{4\pi}\cdot \left( 1+O\left(\frac1{n^2}\right) \right)
=\frac1{4\pi}+O\left(\frac1{n^2}\right) \tag3
\end{align}
$$ as wanted.
