적분 수열 합

$\displaystyle a _{n} = \int _{0} ^{1} ( x ^{n+1} - x ^{n} ) dx$이므로 $\displaystyle \sum\limits _{n=1} ^{n} a _{n}$ $\displaystyle = \int _{0} ^{1} ( x ^{2} - x ) dx + \int _{0} ^{1} ( x ^{3} - x ^{2} ) dx + \cdots + \int _{0} ^{1} ( x ^{11} - x ^{10} ) dx$ $\displaystyle = \int _{0} ^{1} \left\{ ( x ^{2} - x ) + ( x ^{3} - x ^{2} ) + \cdots + ( x ^{11} - x ^{10} ) \right\} dx$ $\displaystyle = \int _{0} ^{1} ( x ^{11} - x ) dx$ $\displaystyle = \left[ \frac{x ^{12}}{12} - \frac{x ^{2}}{2} \right] _{0} ^{1}$$\displaystyle = - \frac{5}{12}$ [다른 풀이] $a _{n}$$\displaystyle = \int _{0} ^{1} ( x ^{n+1} - x ^{n} ) dx$ $\displaystyle = \left[ \frac{x ^{n+2}}{n + 2} - \frac{x ^{n+1}}{n + 1} \right] _{0} ^{1}$ $\displaystyle = \frac{1}{n + 2} - \frac{1}{n + 1}$ ∴ $\displaystyle \sum\limits _{n=1} ^{10} a _{n}$$\displaystyle = \sum\limits _{n=1} ^{10} \left( \frac{1}{n + 2} - \frac{1}{n + 1} \right)$ $\displaystyle = \left( \frac{1}{3} - \frac{1}{2} \right) + \left( \frac{1}{4} - \frac{1}{3} \right) + \cdots + \left( \frac{1}{12} - \frac{1}{11} \right)$ $\displaystyle = \frac{1}{12} - \frac{1}{2}$ $\displaystyle = - \frac{5}{12}$