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Determine the $n^{th}$ partial sum of the series $\sum_{n=1}^{\infty }\frac{1}{n\left ( n+1 \right )}$ and identify the series as convergent or divergent.

$S_{n}=\cfrac{1}{n^{2}+n}$, convergent

$S_{n}=\cfrac{1}{n^{2}+n}$, divergent

$S_{n}=\cfrac{n}{n+1}$, convergent

$S_{n}=\cfrac{n}{n+1}$, divergent