Express $3.72444\ldots$ as a fraction using the formula for geometric progressions I did the following:
$$3.72\overline{4} = 3.724+\left(\left(\frac{4}{10^4}\right)+\left(\frac{4}{10^5}\right)+\left(\frac{4}{10^6}\right)+\cdots\right)$$ 
where $a=3.724$ and $r=\dfrac{1}{10}$
Using $S=\dfrac{a}{1-r}$ I get $\dfrac{37.24}{9}\neq3.72\overline{4}$
The answer given by the text is $\dfrac{838}{225}$
I've been staring at this for an hour and can't figure out where I'm going wrong.

Start your geometric series after $3.72$:
$$
3.72\overline{4} = 3.72+\left(\left(\frac{4}{10^3}\right)+\left(\frac{4}{10^4}\right)+\left(\frac{4}{10^5}\right)+\cdots\right)
$$
Then you have $a=\dfrac{4}{10^3}$ and $r=\dfrac 1{10}$.
The ultimate sum is then
$$
3.72 + \frac{a}{1-r}.
$$
