Answer
$\frac{5}{9}$
Work Step by Step
$0.5555\ldots $
This can be written as,
$0.5555\ldots =0.5+0.05+0.005+0.0005+\cdots $
This is an infinite geometric series:
${{a}_{1}}=0.5$ and ${{a}_{2}}=0.05$.
So, the value of $\left| r \right|$ is,
$\begin{align}
& \left| r \right|=\left| \frac{0.05}{0.5} \right| \\
& =\left| 0.1 \right| \\
& =0.1
\end{align}$
Find the limit of the infinite geometric series:
${{S}_{\infty }}=\frac{{{a}_{1}}}{1-r}$.
$\begin{align}
& {{S}_{\infty }}=\frac{{{a}_{1}}}{1-r} \\
& =\frac{0.5}{1-0.1} \\
& =\frac{0.5}{0.9} \\
& =\frac{5}{9}
\end{align}$
Thus, the fraction notation of the decimal number $0.5555\ldots $ is $\frac{5}{9}$.
