시그마와 부분분수

[출제의도] 합의 기호의 성질을 이해할 수 있는가를 묻는 문제이다. $\displaystyle \sum\limits _{k = 1} ^{9} \left\{ \frac{k ( k + 1 )}{10} + \frac{10}{k ( k + 1 )} \right\}$ $\displaystyle = \frac{1}{10} \sum\limits _{k = 1} ^{9} ( k ^{2} + k ) + 10 \sum\limits _{k = 1} ^{9} \left( \frac{1}{k} - \frac{1}{k + 1} \right)$ $\displaystyle \left. = \frac{1}{10} \sum\limits _{k = 1} ^{9} k ^{2} + \frac{1}{10} \sum\limits _{k = 1} ^{9} k \right.$$\displaystyle \left. + 10 \left\{ \left( \frac{1}{1} - \frac{1}{2} \right) + \left( \frac{1}{2} - \frac{1}{3} \right) + \cdots + \left( \frac{1}{9} - \frac{1}{10} \right) \right\} \right.$ $\displaystyle \left. = \frac{1}{10} \times \frac{9 \cdot 10 \cdot 19}{6} + \frac{1}{10} \times \frac{9 \cdot 10}{2} + 10 \left( 1 - \frac{1}{10} \right) \right.$ $\displaystyle = \frac{57}{2} + \frac{9}{2} + 9$ $= 42$