두 직선 사이 선분 급수

조건을 이용하여 점 ${\mathrm{A}} _{n}$, ${\mathrm{B}} _{n}$의 좌표를 구하면 $\mathrm{A} _{1} ( 1$, $2 )$, $\mathrm{B} _{1} ( 4$, $2 )$, $\mathrm{A} _{2} ( 2$, $4 )$, $\mathrm{B} _{2} ( 8$, $4 )$, $\mathrm{A} _{3} ( 4$, $8 )$, $\mathrm{B} _{3} ( 16$, $8 )$, $\mathrm{A} _{4} ( 8$, $16 )$, $\mathrm{B} _{4} ( 32$, $16 )$, … 이 때, $\displaystyle {\overline{\mathrm{A} \mathit{_{n}} \mathrm{B} \mathit{_{n}}}}$의 길이는 점 ${\mathrm{B}} _{n}$의 $x$좌표에서 점 ${\mathrm{A}} _{n}$의 $x$좌표를 뺀 값이므로 $\displaystyle \sum\limits _{n = 1} ^{\infty} \frac{\mathrm{1}}{{\overline{A _{\mathit{n}} \mathrm{B} _{\mathit{n}}}}}$$\displaystyle = \frac{\mathrm{1}}{{\overline{A _{1} B _{1}}}} + \frac{1}{{\overline{A _{2} B _{2}}}} + \frac{1}{{\overline{A _{3} B _{3}}}} + \cdots$ $\displaystyle = \frac{1}{3} + \frac{1}{6} + \frac{1}{12} + \frac{1}{24} + \cdots$ $\displaystyle = \frac{1}{3} + \frac{1}{3} \cdot \frac{1}{2} + \frac{1}{3} \cdot \frac{1}{4} + \frac{1}{3} \cdot \frac{1}{8} + \cdots$ $\displaystyle = \frac{1}{3} + \frac{1}{3} \cdot \left( \frac{1}{2} \right) + \frac{1}{3} \cdot \left( \frac{1}{2} \right) ^{2}$$\displaystyle + \frac{1}{3} \cdot \left( \frac{1}{2} \right) ^{3} + \cdots$ $\displaystyle = \frac{\frac{1}{3}}{1 - \frac{1}{2}} = \frac{2}{3}$