분모 유리화 급수 합

원 $x ^{2} + y ^{2} = n$이 직선 $\displaystyle y = \sqrt{3} x$와 제$\mathrm{1}$사분면에서 만나는 점의 좌표를 $\left( x _{n} , y _{n} \right)$이라 하면 $x _{n} \mathit{^{2}} + y _{n} \mathit{^{2}} = n$, $\displaystyle y _{n} = \sqrt{3} x _{n}$ $\displaystyle x _{n} \mathit{^{2}} + \left( \sqrt{3} x _{n} \right) ^{2} = n$, $\displaystyle x _{n} \mathit{^{2}} = \frac{n}{4}$ $x _{n} > 0$이므로 $\displaystyle x _{n} = \frac{\sqrt{n}}{2}$ $\displaystyle \sum\limits _{k = 1} ^{80} \frac{1}{x _{k} + x _{k + 1}}$$\displaystyle = \sum\limits _{k = 1} ^{80} \frac{1}{\frac{\sqrt{k}}{2} + \frac{\sqrt{k + 1}}{2}}$ $\displaystyle = \sum\limits _{k = 1} ^{80} \frac{2}{\sqrt{k} + \sqrt{k + 1}}$$\displaystyle = \sum\limits _{k = 1} ^{80} \frac{2 \left( \sqrt{k + 1} - \sqrt{k} \right)}{\left( k + 1 \right) - k}$ $\displaystyle = 2 \sum\limits _{k = 1} ^{80} \left( \sqrt{k + 1} - \sqrt{k} \right)$ $\displaystyle = 2 \left( \sqrt{81} - 1 \right) = 2 \times 8 = 16$