Explanation

(a)
Let $s_{n} = \frac{1}{1.3} + \frac{1}{3.5} + \frac{1}{5.7} + ... + \frac{1}{(2n - 1)(2n + 1)}$
$= \frac{1}{2}\left\lbrack \left( 1 - \frac{1}{3} \right) + \left( \frac{1}{3} - \frac{1}{5} \right) + ... + \left( \frac{1}{2n - 1} - \frac{1}{2n + 1} \right) \right\rbrack$
$= \frac{1}{2}\left\lbrack 1 - \frac{1}{2n + 1} \right\rbrack$
$\therefore\ \ \ \ \ \ \ \lim_{n \rightarrow \infty}\ \ s_{n} = \lim_{n \rightarrow \infty}\ \frac{1}{2}\left\lbrack 1 - \frac{1}{2n + 1} \right\rbrack = \frac{1}{2}$