부분분수 무한급수

수열의 일반항을 $a _{n}$, $n$항까지의 합을 $S _{n}$이라 하면 $\displaystyle a _{n} = {\frac{n}{( n ^{2} - n + 1 ) ( n ^{2} + n + 1 )}}$ $\displaystyle S _{n} = \sum\limits _{k = 1} ^{n} \frac{1}{2} \left( \frac{1}{k ^{2} - k + 1} - \frac{1}{k ^{2} + k + 1} \right)$ $\displaystyle = \frac{1}{2} \left\{ \left( 1 - \frac{1}{3} \right) + \left( \frac{1}{3} - \frac{1}{7} \right) + \left( \frac{1}{7} - \frac{1}{13} \right) + \right.$ $\displaystyle \left. \cdots + \left( \frac{1}{n ^{2} - n + 1} - \frac{1}{n ^{2} + n + 1} \right) \right\}$ $\displaystyle = \frac{1}{2} \left( 1 - \frac{1}{n ^{2} + n + 1} \right)$ $\therefore$$\displaystyle \lim\limits _{n \rightarrow \infty} {S _{n} = \frac{1}{2}}$