등차수열의 세 항 곱의 역수의 무한합

등차수열 $\left\{ a _{n} \right\}$의 공차를 $d$라 하면 $a _{10} - a _{2}$$= 8 d$$= 4$ ∴ $\displaystyle d = \frac{1}{2}$ ∴ $\displaystyle a _{n} = \frac{n + 5}{2}$ (∵ $a _{1} = 3$) ∴ $\displaystyle \sum\limits _{n=1} ^{\infty} \frac{1}{a _{n} a _{n+1} a _{n+2}}$ $\displaystyle = \lim\limits _{n\rightarrow \infty} \sum\limits _{k = 1} ^{n} \frac{1}{a _{k} a _{k+1} a _{k+2}}$ $\displaystyle = \lim\limits _{n\rightarrow \infty} \sum\limits _{k = 1} ^{n} \frac{8}{( k + 5 ) ( k + 6 ) ( k + 7 )}$ $\displaystyle = \lim\limits _{n\rightarrow \infty} \sum\limits _{k=1} ^{n} 4 \left( \frac{1}{( k + 5 ) ( k + 6 )} - \frac{1}{( k + 6 ) ( k + 7 )} \right)$ $\displaystyle = \lim\limits _{n\rightarrow \infty} 4 \left( \frac{1}{42} - \frac{1}{( n + 6 ) ( n + 7 )} \right)$ $\displaystyle = \frac{2}{21}$