Explanation

(c)
Let $S = 1 + \frac{2 \bullet 1}{3 \bullet 2} + \frac{2 \bullet 5}{3 \bullet 6}\left( \frac{1}{2} \right)^{2} + \frac{2 \bullet 5 \bullet 8}{3 \bullet 6 \bullet 9}\left( \frac{1}{2} \right)^{3} + ...$
$= 1 + \frac{\frac{2}{3}}{1}\left( \frac{1}{2} \right) + \frac{\left( \frac{2}{3} \right)\left( \frac{5}{3} \right)}{2!}\left( \frac{1}{2} \right)^{2} + \frac{\left( \frac{2}{3} \right)\left( \frac{5}{3} \right)\left( \frac{8}{3} \right)}{3!}\left( \frac{1}{2} \right)^{3} + ...$
$= \left( 1 - \frac{1}{2} \right)^{\frac{- 2}{3}} = \left( \frac{1}{2} \right)^{\frac{- 2}{3}} = 2^{\frac{2}{3}} = 4^{\frac{1}{3}}$
$\left\lbrack \because{(1 - x)}^{- n} = 1 + nx + \frac{n(n + 1)}{2!}x^{2} + ... \right\rbrack$