$S = \frac{1}{3.5} + \frac{1}{5.7} +...+ \frac{1}{2005.2007}$
$= \frac{1}{2}.(\frac{2}{3.5} + \frac{2}{5.7} +...+ \frac{2}{2005.2007})$
$= \frac{1}{2}.(\frac{1}{3} - \frac{1}{5} + \frac{1}{5} - \frac{1}{7} +...+ \frac{1}{2005} - \frac{1}{2007})$
$= \frac{1}{2}.(\frac{1}{3} - \frac{1}{2007})$
$= \frac{1}{2}.\frac{668}{2007} = \frac{334}{2007}$