회전하는 점들 사이 거리의 무한급수

$\displaystyle \overline{{\mathrm{OA}} _{n}}$$\displaystyle = \left( \frac{3}{4} \right) ^{n}$이고 $\angle$${\mathrm{A}} _{n{\mathrm{O}}} {\mathrm{A}} _{n+1}$$\displaystyle = \frac{\pi}{3}$이므로 삼각형 ${\mathrm{OA}} _{n} {\mathrm{A}} _{n+1}$에서 코사인정리를 이용하면 $\displaystyle {\overline{{\mathrm{A}} _{n} {\mathrm{A}} _{n+1}}} ^{2}$$\displaystyle = \left( \frac{3}{4} \right) ^{2n} + \left( \frac{3}{4} \right) ^{2n+2} - 2 \left( \frac{3}{4} \right) ^{2n+1} \cos \frac{\pi}{3}$$\displaystyle = \frac{13}{16} \left( \frac{3}{4} \right) ^{2n}$ ∴ $\displaystyle \overline{{\mathrm{A}} _{n} {\mathrm{A}} _{n+1}}$$\displaystyle = \frac{\sqrt{13}}{4} \left( \frac{3}{4} \right) ^{n}$ $\displaystyle \sum\limits _{n = 1} ^{\infty} \overline{\mathrm{A} \mathit{\mathrm{_{n}}} \mathrm{A} \mathit{\mathrm{_{n + 1}}}}$$\displaystyle = \sum\limits _{n=1} ^{\infty} \frac{\sqrt{13}}{4} \left( \frac{3}{4} \right) ^{n}$$\displaystyle = \frac{\frac{3 \sqrt{13}}{16}}{1 - \frac{3}{4}}$$\displaystyle = \frac{3 \sqrt{13}}{4}$