곡선과 직선 사이 넓이의 급수합

곡선 $y = x ^{n}$ 과 직선 $y = 1 ,$ $y$축으로 둘러싸인 넓이 $S _{n}$은 $\displaystyle S _{n} = 1 - \int _{0} ^{1} {x ^{n} dx} = 1 - \left[ \frac{1}{n + 1} x ^{n+1} \right] _{0} ^{1} = \frac{n}{n + 1}$ 따라서, $\displaystyle \sum\limits _{n = 1} ^{\infty} {\frac{S _{n}}{n ^{2}}} = \sum\limits _{n = 1} ^{\infty} \frac{\frac{n}{n + 1}}{n ^{2}} = \sum\limits _{n = 1} ^{\infty} \frac{1}{n ( n + 1 )}$ $\displaystyle = \lim\limits _{n \rightarrow \infty} {\sum\limits _{k = 1} ^{n} {\frac{1}{k ( k + 1 )}}}$ $\displaystyle = \lim\limits _{n \rightarrow \infty} {\left\{ ( 1 - \frac{1}{2} ) + ( \frac{1}{2} - \frac{1}{3} ) + \cdots + ( \frac{1}{n} - \frac{1}{n + 1} ) \right\}}$ $\displaystyle = \lim\limits _{n \rightarrow \infty} {\left( 1 - \frac{1}{n + 1} \right)}$ $= 1$