Answer
$\frac{416909}{99900}$
Work Step by Step
Convert repeating part $326$ into a series.
$4.17326326326 . . .=(417+ \Sigma _{n=1}^{\infty}\frac{326}{(1000)^{n}})10^{-2}$
$=(417+ \Sigma _{n=1}^{\infty}\frac{326}{1000}\frac{1}{(1000)^{n-1}})10^{-2}$
The sum of a geometric series $\Sigma _{n=1}^{\infty}ar^{n-1}$ equals $a/1-r$
Thus,
$=(417+\frac{\frac{326}{1000}}{1-\frac{1}{1000}})10^{-2}$
$=(417+\frac{326}{999})10^{-2}$
$=(\frac{416583+326}{999})10^{-2}$
$=\frac{416909}{99900}$
