Sum of $\frac{1}{n^2}$ = $\pi$ !!

Relevant wiki: Partial fractions , Telescoping series

$\begin{aligned} S&=\displaystyle\sum_{n=1}^{\infty}\dfrac{1}{n^2+2n}\\ &=\displaystyle\sum_{n=1}^{\infty}\dfrac{1}{n(n+2)}\\ &=\dfrac{1}{2}\displaystyle\sum_{n=1}^{\infty}\left[\dfrac{1}{n}-\dfrac{1}{n+2}\right]\phantom{ccccccccc} \color{#3D99F6}\text{by partial fractions}\\ &=\dfrac{1}{2}\left(1+\dfrac{1}{2}\right)=\boxed{\dfrac{3}{4}}\phantom{ccccccccc} \color{#3D99F6}\text{by Telescoping series sum} \end{aligned}$