사인곡선 넓이 급수

$\displaystyle S _{n} = \int _{(n-1)\pi} ^{n \pi} \left| \left( \frac{1}{2} \right) ^{n} \sin x \right| dx$ $\displaystyle = \left( \frac{1}{2} \right) ^{n} \int _{(n-1)\pi} ^{n \pi} \left| \sin x \right| dx$ 이 때, $y = | \sin x |$는 주기가 $\pi$인 주기함수이므로 임의의 자연수 $n$에 대하여 $\displaystyle \int _{(n-1)\pi} ^{n \pi} \left| \sin x \right| dx = \int _{0} ^{\pi} \sin x dx$ $= \left[ - \cos x \right] _{0} ^{\pi}$$= 2$ ∴ $\displaystyle S _{n} = \left( \frac{1}{2} \right) ^{n} \int _{(n-1)\pi} ^{n \pi} \left| \sin x \right| dx$ $\displaystyle = 2 \cdot \left( \frac{1}{2} \right) ^{n}$ $\displaystyle = \left( \frac{1}{2} \right) ^{n-1}$ ∴ $\displaystyle \sum\limits _{n=1} ^{\infty} S _{n} = \sum\limits _{n=1} ^{\infty} \left( \frac{1}{2} \right) ^{n-1}$$\displaystyle = \frac{1}{1 - \frac{1}{2}}$$= 2$ ∴ $\alpha = 2$ ∴ $50 \alpha = 100$