Answer
$$5$$
Work Step by Step
$$\eqalign{
& 1 + 0.8 + 0.64 + 0.512 + \cdots \cr
& {\text{Sum of a Geometric Series }}\left( {{\text{see page 635}}} \right) \cr
& {\text{The infinite geometric series}} \cr
& a + ar + a{r^2} + a{r^3} + \cdots \cr
& {\text{converges}}{\text{, if }}r{\text{ is in }}\left( { - 1,1} \right),{\text{ to the sum }}\frac{a}{{1 - r}}.{\text{ And diverges if }}r{\text{ is not in }}\left( { - 1,1} \right) \cr
& {\text{then this is a geometric series}}{\text{, with }}a = {a_1} = 1 \cr
& r = \frac{{0.8}}{1} = 0.8 \cr
& {\text{Since }}r{\text{ is in the interval }}\left( { - 1,1} \right),{\text{ the series converges and has a sum }}\frac{a}{{1 - r}} \cr
& \frac{a}{{1 - r}} = \frac{1}{{1 - 0.8}} \cr
& {\text{simplifying}} \cr
& \frac{a}{{1 - r}} = 5 \cr} $$
