나열된 수의 합

나열된 모든 수의 합은 $\begin{aligned} ( 9 + 0.9 + 0.09 + 0.009 + 0.0009 + {\cdots} ) \\ \\ + ( 0.9 + 0.09 + 0.009 + 0.0009 + {\cdots} ) \\ \\ + ( 0.09 + 0.009 + 0.0009 + {\cdots} ) \\ \\ + ( 0.009 + 0.0009 + {\cdots} ) \\ \\ + {\cdots} \end{aligned}$ $\displaystyle = 10 + 1 + \frac{1}{10} + \frac{1}{100} + \cdots$ $\displaystyle = \frac{10}{1 - \frac{1}{10}} = \frac{100}{9}$ (별해) $S = 9 + 2 \times 0.9 + 3 \times 0.09 + 4 \times 0.009 + \cdots$ …① $0.1 S = 0.9 + 2 \times 0.09 + 3 \times 0.009 + \cdots$ … ② ①$-$ ②에서 $0.9 S = 9 + 0.9 + 0.09 + 0.009 + \cdots$ $\displaystyle = \frac{9}{1 - 0.1} = 10$ $\displaystyle \therefore S = \frac{100}{9}$