Answer
Converges
Work Step by Step
$$\sum ^{\infty }_{n=1}\left( \dfrac {1}{n+1}-\dfrac {1}{n+2}\right) =\left( \dfrac {1}{1+1}-\dfrac {1}{1+2}\right) +\left( \dfrac {1}{2+1}-\dfrac {1}{2+2}\right) \ldots +\left( \dfrac {1}{n+1}-\dfrac {1}{n+2}\right) =\dfrac {1}{2}-\dfrac {1}{n+2}=\dfrac {1}{2}$$
So the series converges