Answer
$\dfrac{1}{9}$
Work Step by Step
Sum of the first $n$ terms terms of a geometric sequence can be calculated as: $s_n=\dfrac{a_1}{(1-r)}$
$r$ refers to common ratio and $r=\dfrac{1}{10}=0.1$ and $S_n=0.\overline{1}$
Apply the formula.
$s_n=\dfrac{0.1}{(1-0.1)}=\dfrac{0.1}{0.9}=\dfrac{1}{9}$