That is a collapsing sum: $\displaystyle \sum\limits_{n = 1}^\infty {\left( {\frac{1}{{(n + 1)(n + 2)}}} \right)} = \sum\limits_{n = 1}^\infty {\left( {\frac{1}{{(n + 1)}} - \frac{1}{{(n + 2)}}} \right)} $
Any partial sum looks like $\displaystyle {S_K} = \sum\limits_{n = 1}^N {\left( {\frac{1}{{(n + 1)}} - \frac{1}{{(n + 2)}}} \right)} = \frac{1}{2} - \frac{1}{{N + 2}}$ thus $\displaystyle \left(S_N\right)\to~?$.
Do you know what it actually means for a infinite series to converge?
What do partial sums have to do with?
Do you understand how $\displaystyle {S_N} = \sum\limits_{n = 1}^N {\left( {\frac{1}{{(n + 1)}} - \frac{1}{{(n + 2)}}} \right)} = \frac{1}{2} - \frac{1}{{N + 2}}$ actually works?
If not, there is no point in your trying the question.