Answer
$${S_\infty } = - \frac{9}{{20}}$$
Work Step by Step
$$\eqalign{
& - \frac{3}{4} + \frac{1}{2} - \frac{1}{3} + \frac{2}{9} + \cdots \cr
& \frac{{1/2}}{{ - 3/4}} = - \frac{2}{3},\,\,\,\,\frac{{ - 1/3}}{{1/2}} = - \frac{2}{3} \cr
& {\text{This is a geometric series with }}{a_1} = - \frac{3}{4}{\text{ and }}r = - \frac{2}{3} \cr
& {\text{The sum of the series is }}{S_\infty } = \frac{{{a_1}}}{{1 - r}} \cr
& {S_\infty } = \frac{{ - 3/4}}{{1 - \left( { - 2/3} \right)}} \cr
& {\text{Simplifying}} \cr
& {S_\infty } = - \frac{9}{{20}} \cr} $$
