정삼각형 중점 반복 각

$\mathrm{A} _{n} ( x _{n} , y _{n} )$이라 두면 $\displaystyle \lim\limits _{n \rightarrow \infty} {} \tan \left( \angle AOA _{n} \right) = \lim\limits _{n \rightarrow \infty} {} \frac{y _{n}}{x _{n}}$이므로 아래 그림에서

삼각형의 한 변의 길이를 $a = 12$라 두면 ⅰ) $\displaystyle \begin{aligned} & \therefore \lim\limits _{n \rightarrow \infty} {} x _{n} \end{aligned}$ $\displaystyle = a - \frac{1}{2} a \times \cos \frac{\pi}{3} - \left( \frac{1}{2} \right) ^{2} a \times \cos \frac{\pi}{3} + \left( \frac{1}{2} \right) ^{3} \times a$ $\displaystyle - \left( \frac{1}{2} \right) ^{4} a \times \cos \frac{\pi}{3} - \left( \frac{1}{2} \right) ^{5} a \times \cos \frac{\pi}{3} + \left( \frac{1}{2} \right) ^{3} \times a + \cdots$ $\displaystyle = a \times \left\{ 1 + \left( \frac{1}{2} \right) ^{3} + \left( \frac{1}{2} \right) ^{6} + \cdots \right\}$ $\displaystyle - \left( \frac{1}{2} \right) a \times \cos \frac{\pi}{3} \times \left\{ 1 + \left( \frac{1}{2} \right) ^{3} + \left( \frac{1}{2} \right) ^{6} + \cdots \right\}$$\displaystyle - \left( \frac{1}{2} \right) ^{2} a \times \cos \frac{\pi}{3} \times \left\{ 1 + \left( \frac{1}{2} \right) ^{3} + \left( \frac{1}{2} \right) ^{6} + \cdots \right\}$ $\displaystyle \begin{aligned} & = a \times \left( 1 - \frac{1}{4} - \frac{1}{8} \right) \times \left\{ 1 + \left( \frac{1}{2} \right) ^{3} + \left( \frac{1}{2} \right) ^{6} + \cdots \right\} \end{aligned}$ $\displaystyle = a \times \frac{5}{8} \times \frac{1}{1 - \left( \frac{1}{2} \right) ^{3}} = \frac{5}{7} a$ ⅱ) $\displaystyle \begin{aligned} & \therefore \lim\limits _{n \rightarrow \infty} {} y _{n} \end{aligned}$ $\displaystyle = 0 + \frac{1}{2} a \times \sin \frac{\pi}{3} - \left( \frac{1}{2} \right) ^{2} a \times \sin \frac{\pi}{3} + \left( \frac{1}{2} \right) ^{4} a \times \sin \frac{\pi}{3}$$\displaystyle - \left( \frac{1}{2} \right) ^{5} a \times \sin \frac{\pi}{3} + \left( \frac{1}{2} \right) ^{7} a \times \sin \frac{\pi}{3} + \cdots$ $\displaystyle = a \times \sin \frac{\pi}{3} \times \left[ \left\{ \frac{1}{2} + \left( \frac{1}{2} \right) ^{4} + \left( \frac{1}{2} \right) ^{7} + \cdots \right\} - \left\{ \left( \frac{1}{2} \right) ^{2} + \left( \frac{1}{2} \right) ^{5} + \left( \frac{1}{2} \right) ^{8} \cdots \right\} \right]$ $\displaystyle = \frac{\sqrt{3}}{2} a \times \left\{ \frac{\frac{1}{2}}{1 - \left( \frac{1}{2} \right) ^{3}} - \frac{\left( \frac{1}{2} \right) ^{2}}{1 - \left( \frac{1}{2} \right) ^{3}} \right\} = \frac{\sqrt{3}}{7} a$ $\displaystyle \therefore \lim\limits _{n \rightarrow \infty} \tan \left( \angle \mathrm{A} OA _{n} \right) = \lim\limits _{n \rightarrow \infty} \frac{y _{n}}{x _{n}} = \frac{\frac{\sqrt{3}}{7} a}{\frac{5}{7} a} = \frac{\sqrt{3}}{5}$