How to evaluate a fractional series

How to evaluate the series 1/(r+2)(r+3) for the first n terms.

[angelmendez-rivera]

There is actually the much simpler way of decomposing 1/[(r + 2)·(r + 3)] = (3 – 2)/[(r + 2)·(r + 3)] = [(r + 3) – (r + 2)]/[(r + 2)·(r + 3)] = 1/(r + 2) – 1/(r + 3), resulting in the telescoping summation. Also, for most purposes, leaving it as 1/3 – 1/(n + 3) is better, mainly because when you evaluate the limit as n —> ♾, you can trivially simplify this to 1/3.