포물선 조각 넓이 급수

$\displaystyle - \frac{1}{4 ^{n}} ( x - n ) ( x - n - 1 )$에 $n = 0$을 대입하면 $- x ( x - 1 )$이므로 음이 아닌 정수 $n$에 대하여 $f ( x )$$\displaystyle = - \frac{1}{4 ^{n}} ( x - n ) ( x - n - 1 )$ ($n \leq x < n + 1$) 이다. 이때, $\displaystyle a _{n} = \int _{n} ^{n+1} f ( x ) dx$ 이라 하면 $S _{n}$$\displaystyle = \int _{0} ^{n+1} f ( x ) dx$ $\displaystyle = \int _{0} ^{1} f ( x ) dx + \int _{1} ^{2} f ( x ) dx + \cdots + \int _{n} ^{n+1} f ( x ) dx$ $= a _{0} + a _{1} + \cdots + a _{n}$ $\displaystyle = \sum\limits _{i=0} ^{n} a _{i}$ 그런데 $a _{n}$$\displaystyle = \int _{n} ^{n+1} \left\{ - \frac{1}{4 ^{n}} ( x - n ) ( x - n - 1 ) \right\} dx$ $\displaystyle = \frac{1}{6} \cdot \frac{1}{4 ^{n}}$ ∴ $\displaystyle \left( \int _{\alpha} ^{\beta} a ( x - \alpha ) ( x - \beta ) dx = - \frac{a}{6} ( \beta - \alpha ) ^{3} \right)$ 이므로 $\displaystyle \lim\limits _{n\rightarrow \infty} S _{n}$$\displaystyle = \sum\limits _{n=0} ^{\infty} a _{n}$$\displaystyle = \sum\limits _{n=0} ^{\infty} \frac{1}{6} \cdot \frac{1}{4 ^{n}}$$\displaystyle = \frac{\frac{1}{6}}{1 - \frac{1}{4}}$$\displaystyle = \frac{2}{9}$