Answer
$${S_\infty } = 1$$
Work Step by Step
$$\eqalign{
& 0.9 + 0.09 + 0.009 + 0.0009 + \cdots \cr
& \frac{{0.09}}{{0.9}} = \frac{1}{{10}},\,\,\,\,\frac{{0.009}}{{0.09}} = \frac{1}{{10}} \cr
& {\text{This is a geometric series with }}{a_1} = 0.9{\text{ and }}r = \frac{1}{{10}} = 0.1 \cr
& {\text{The sum of the series is }}{S_\infty } = \frac{{{a_1}}}{{1 - r}}\,\,\left( {{\text{where }}\left| r \right| < 1} \right) \cr
& {S_\infty } = \frac{{0.9}}{{1 - 0.1}} \cr
& {S_\infty } = 1 \cr} $$
