Jun 8, 2021 - 07:04

An infinite series problem.

By Vamshi Jandhyala in mathematics

June 12, 2021

Power

Solution

We have

$$ \begin{aligned} \sum_{n=1}^\infty \frac{1}{F_{n} F_{n+2}} &= \sum_{n=1}^\infty \frac{1}{F_{n+1}} \left( \frac{1}{F_{n}} - \frac{1}{F_{n+2}} \right) \\
&= \frac{1}{F_1 F_2} - \frac{1}{F_2 F_3} + \frac{1}{F_2 F_3} - \frac{1}{F_3 F_4} + \dots \\
&= \frac{1}{F_1F_2} = 1 \end{aligned} $$

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