# Jun 8, 2021 - 07:04

#### An infinite series problem.

By Vamshi Jandhyala in mathematics

June 12, 2021

## Power

— 級数bot (@infseriesbot) June 8, 2021

## Solution

We have

$$
\begin{aligned}
\sum_{n=1}^\infty \frac{1}{F_{n} F_{n+2}} &= \sum_{n=1}^\infty \frac{1}{F_{n+1}} \left( \frac{1}{F_{n}} - \frac{1}{F_{n+2}} \right) \\

&= \frac{1}{F_1 F_2} - \frac{1}{F_2 F_3} + \frac{1}{F_2 F_3} - \frac{1}{F_3 F_4} + \dots \\

&= \frac{1}{F_1F_2} = 1
\end{aligned}
$$