Explanation

(b)
Let $S = \frac{1}{1.2} + \frac{1.3}{1.2.3.4} + \frac{1.3.5}{1.2.3.4.5.6} + ...$
$\therefore T_{n} = \frac{1.3.5.\ldots.(2n - 1)}{1.2.3.\ldots.(2n - 1)2n} \times \frac{(2.4.8.\ldots.2n)}{(2.4.8.\ldots.2n)}$
$= \frac{(2n)!}{(2n)!.2^{n}(n!)} = \frac{1}{2^{n}(n!)}$
$\therefore S = \sum_{}^{}{T_{n} = \frac{1}{2.1!} + \frac{1}{2^{2}.2!} + \frac{1}{2^{3}.3!} + ...}$
$= e^{1/2} - 1$