Calculate $S_3,S_{4}$, and $S_{5}$ and then find the sum for the telescoping series $$S=\sum_{n=3}^\infty \left(\frac{1}{n+1}-\frac{1}{n+2}\right)$$ where $S_k$ is the partial sum using the first $k$ values of $n$.

$S_3$ =

$S_{4}$ =

$S_{5}$ =

$S$ =