Answer
$1/9$
Work Step by Step
We add this step wise,
$0.1+0.01=0.11$
$0.11+0.001 = 0.111$
$0.111+0.0001 = 0.1111$
The patern shows us the answer is is $.111111...$
That can be calculated with the infinity sum of a geometric series formula,
$S_n = \frac{a}{1-r}$ Where $a$ is the first term and $r$ is the common ratio.
$0.01/0.1 = 0.1$
The first term is $0.1$
$$S_n = \frac{0.1}{1-0.1} = 1/10* 10/9 = 1/9$$
