Answer
$\frac{2}{9}$
Work Step by Step
The repeating decimal 0.2222222222 . . . can be expressed as the infinite geometric series:
$0.222222..=0.2+0.2(\frac{1}{10})+0.2(\frac{1}{100})^{2}+0.2(\frac{1}{100})^{3}...$
$a=0.2$
$r=\frac{1}{10}$
$\Sigma=\frac{0.2}{1-\frac{1}{10}}=\frac{2}{9}$
and $\frac{2}{9}$ is also $0.22222222...$
