피보나치형 급수

$a _{n + 2} = a _{n + 1} + a _{n}$ $\rightarrow a _{n} = a _{n+2} - a _{n+1}$에서 $\displaystyle \frac{a _{n}}{a _{n+1} a _{n+2}}$$\displaystyle = \frac{a _{n+2} - a _{n+1}}{a _{n+1} a _{n+2}}$$\displaystyle = \frac{1}{a _{n+1}} - \frac{1}{a _{n+2}}$이므로 $\displaystyle \sum\limits _{n=1} ^{n} \frac{a _{k}}{a _{k+1} a _{k+2}}$$\displaystyle = \sum\limits _{k=1} ^{n} \left( \frac{1}{a _{k+1}} - \frac{1}{a _{k+2}} \right)$ $\displaystyle = \left( \frac{1}{a _{2}} - \frac{1}{a _{3}} \right) + \left( \frac{1}{a _{3}} - \frac{1}{a _{4}} \right) + \left( \frac{1}{a _{4}} - \frac{1}{a _{5}} \right) + \cdots$$\displaystyle + \left( \frac{1}{a _{n+1}} - \frac{1}{a _{n+2}} \right)$ $\displaystyle = \frac{1}{a _{2}} - \frac{1}{a _{n+2}}$$\displaystyle = \frac{1}{2} - \frac{1}{a _{n+2}}$ 수열 $\left\{ a _{n} \right\}$은 $a _{n + 2} = a _{n + 1} + a _{n}$을 만족하므로 증가하는 수열이고 따라서 $\displaystyle \lim\limits _{n\rightarrow \infty} \frac{1}{a _{n}} = 0$이다. $\displaystyle \sum\limits _{n=1} ^{\infty} \frac{a _{n}}{a _{n+1} a _{n+2}} = \frac{1}{2}$