Evaluate: $$\frac{1}{{15}} + \frac{1}{{35}} + \frac{1}{{63}} + \frac{1}{{99}} + \frac{1}{{143}}$$

$$\eqalign{ & \frac{1}{{15}} + \frac{1}{{35}} + \frac{1}{{63}} + \frac{1}{{99}} + \frac{1}{{143}} \cr & = \frac{1}{2}\left[ {\frac{1}{3} - \frac{1}{5} + \frac{1}{5} - \frac{1}{7}\,.\,.\,.\,.\,.\,\frac{1}{{11}} - \frac{1}{{13}}} \right] \cr & = \frac{1}{2}\left[ {\frac{1}{3} - \frac{1}{{13}}} \right] \cr & = \frac{1}{2} \times \frac{{10}}{{39}} \cr & = \frac{5}{{39}} \cr} $$