Answer
$\frac{343}{99}$
Work Step by Step
$3.4646\ldots $
This can be written as,
$3.4646\ldots =3+0.4646\ldots $
And,
$0.464646\ldots =0.46+0.0046+0.000046+\cdots $
This is an infinite geometric series:
So, the value of $\left| r \right|$ is,
$\begin{align}
& \left| r \right|=\left| \frac{0.0046}{0.46} \right| \\
& =\left| 0.01 \right| \\
& =0.01
\end{align}$
Find the limit of the infinite geometric series by using the formula ${{S}_{\infty }}=\frac{{{a}_{1}}}{1-r}$.
$\begin{align}
& {{S}_{\infty }}=\frac{{{a}_{1}}}{1-r} \\
& =\frac{0.46}{1-0.01} \\
& =\frac{0.46}{0.99} \\
& =\frac{46}{99}
\end{align}$
The fraction notation of the decimal number $0.4646\ldots $ is $\frac{46}{99}$.
And the fraction notation of the decimal number $3.4646\ldots $ is:
$3+\frac{46}{99}=\frac{343}{99}$
Thus, the fraction notation of the decimal number $3.4646\ldots $ is $\frac{343}{99}$.
