User:BQH5HnAV8KlwMs/Sandbox

$$ F_1 = 1, F_2 = 1, F_3 = 2, F_4 = 3, F_5 = 5, F_6 = 8, F_7 = 13, \cdots $$

$$ \sum_{n=2}^{\infty} \frac{1}{F_{n-1} \times F_{n+1}} = \lim_{m \to \infty} \sum_{n=2}^{m} \frac{1}{F_{n-1} \times F_{n+1}} $$

$$ = \frac{1}{F_1 \times F_3} + \frac{1}{F_2 \times F_4} + \frac{1}{F_3 \times F_5} + \cdots + \frac{1}{F_{n-1} \times F_{n+1}} + \cdots $$

$$ = \frac{1}{1 \times 2} + \frac{1}{1 \times 3} + \frac{1}{2 \times 5} + \frac{1}{3 \times 8} + \cdots + \frac{1}{F_{n-1} \times F_{n+1}} + \cdots $$

$$ = \frac{1}{1}(\frac{1}{1} - \frac{1}{2}) + \frac{1}{2}(\frac{1}{1} - \frac{1}{3}) + \frac{1}{3}(\frac{1}{2} - \frac{1}{5}) + \frac{1}{5}(\frac{1}{3} - \frac{1}{8}) + \cdots + \frac{1}{F_n}(\frac{1}{F_{n-1}} - \frac{1}{F_{n+1}}) + \cdots $$

$$ = \lim_{m \to \infty} (\frac{1}{F_2}(\frac{1}{F_1} - \frac{1}{F_3}) + \frac{1}{F_3}(\frac{1}{F_2} - \frac{1}{F_4}) + \frac{1}{F_4}(\frac{1}{F_3} - \frac{1}{F_5}) + \frac{1}{F_5}(\frac{1}{F_4} - \frac{1}{F_6}) + \cdots + \frac{1}{F_m}(\frac{1}{F_{m-1}} - \frac{1}{F_{m+1}})) $$

$$ = \lim_{m \to \infty} (\frac{1}{F_1} \times \frac{1}{F_2} - \frac{1}{F_2} \times \frac{1}{F_3} + \frac{1}{F_2} \times \frac{1}{F_3} - \frac{1}{F_3} \times \frac{1}{F_4} + \frac{1}{F_3} \times \frac{1}{F_4} - \cdots - \frac{1}{F_m} \times \frac{1}{F_{m+1}}) $$

$$ = \lim_{m \to \infty} (\frac{1}{F_1} \times \frac{1}{F_2} - \frac{1}{F_m} \times \frac{1}{F_{m+1}}) $$

$$ = 1 $$