b

$$\frac{p}{q}=0.1230\overline{443}$$

Bearbeitung §

$$\begin{array}{rl}0.1230\overline{443}& =0.123+0.0000443+0.0000000443+0.0000000000443+\cdots \\ & =0.123+\sum _{k=0}^{\infty }0.0001\cdot 0.001^k\cdot 0.443\\ & =0.123+0.0001\cdot 0.443\cdot \sum _{k=0}^{\infty }0.001^k\\ & =0.123+0.0000443\cdot \frac{1}{1-0.001}\\ & =0.123+\frac{0.0000443}{0.999}\\ & =\frac{123}{1000}+\frac{443}{9990000}\\ & =\frac{1228770}{9990000}+\frac{443}{9990000}\\ & =\frac{1229213}{9990000}\\ & ⟹p=1229213,q=9990000\end{array}$$