a

$$\frac{p}{q}=0.\overline{4711}$$

Bearbeitung §

$$\begin{array}{rl}0.\overline{4711}& =0.4711+0.00004711+\cdots \\ & =\sum _{k=0}^{\infty }0.4711\cdot \frac{1}{10000^k}\\ & =0.4711\cdot \sum _{k=0}^{\infty }(\frac{1}{10^4}{)}^k\\ & =0.4711\cdot \frac{1}{1-\frac{1}{10^4}}\\ & =0.4711\cdot \frac{1}{\frac{9999}{10000}}\\ & =0.4711\cdot \frac{10000}{9999}\\ & =\frac{4711}{9999}\\ ⟹& p=4711,q=9999\end{array}$$