Answer
$${S_\infty } = \frac{4}{3}$$
Work Step by Step
$$\eqalign{
& \sum\limits_{i = 1}^\infty {{{\left( {\frac{4}{7}} \right)}^i}} \cr
& = \sum\limits_{i = 1}^\infty {\frac{4}{7}{{\left( {\frac{4}{7}} \right)}^{i - 1}}} \cr
& {\text{The }}n{\text{th term is }}{a_n} = {a_1}{r^{n - 1}},\, \cr
& {a_n} = \frac{4}{7}{\left( {\frac{4}{7}} \right)^{i - 1}}{\text{ then, }}{a_1} = \frac{4}{7}{\text{ and }}r = \frac{4}{7} \cr
& {\text{The sum of the Terms of an Infinite Geometric Sequence is}} \cr
& {S_\infty } = \frac{{{a_1}}}{{1 - r}},{\text{ }}\left( {{\text{where }}\left| r \right| < 1} \right) \cr
& {\text{Then}} \cr
& {S_\infty } = \frac{{4/7}}{{1 - 4/7}} \cr
& {S_\infty } = \frac{4}{3} \cr} $$
