Answer
$\dfrac{26}{111}$
Work Step by Step
We have:
$0.234234234= \dfrac{234}{1000}+\dfrac{234}{(1000)^2}+\dfrac{234}{(1000)^3}+....=\dfrac{234}{1000}(1+ \dfrac{1}{1000}+ \dfrac{1}{(1000)^2}+..)$
The sum of a geometric series $1+ \dfrac{1}{1000}+ \dfrac{1}{(1000)^2}+..$ can be found as:
$S=\dfrac{a}{1-r}$
Here, $a=1, r=\dfrac{1}{1000}=0.001$;
Thus,
$S=\dfrac{234}{1000}(\dfrac{1}{1-0.001})=\dfrac{26}{111}$