Answer
(a) $\frac{1}{f_{n-1}f_{n}}-\frac{1}{f_{n}f_{n+1}}=\frac{1}{f_{n-1}f_{n+1}}$
(b) $\Sigma^{\infty}_{n=2} \frac{1}{f_{n-1}f_{n+1}}=1$
(c) $\Sigma^{\infty}_{n=2} \frac{f_{n}}{f_{n-1}f_{n+1}} = 2$
Work Step by Step
a)
$\frac{1}{f_{n-1}f_{n}}-\frac{1}{f_{n}f_{n+1}}$
$=\frac{f_{n+1}}{f_{n-1}f_{n}f_{n+1}}-\frac{f_{n-1}}{f_{n-1}f_{n}f_{n+1}}$
$=\frac{f_{n+1}-f_{n-1}}{f_{n-1}f_{n}f_{n+1}}$
Given
$f_{n} = f_{n-1}+f_{n-1}$
$f_{n+1} = f_{n} + f_{n-1}$
$f_{n} = f_{n+1} - f_{n-1}$
$=\frac{f_{n}}{f_{n-1}f_{n}f_{n+1}}$
$=\frac{1}{f_{n-1}f_{n+1}}$
b)
from part (a) we know that $\frac{1}{f_{n-1}f_{n+1}} = \frac{1}{f_{n-1}f_{n}} - \frac{1}{f_{n}f_{n+1}}$
$\Sigma^{\infty}_{n=2} \frac{1}{f_{n-1}f_{n+1}} = \Sigma^{\infty}_{n=2} \frac{1}{f_{n-1}f_{n}} - \frac{1}{f_{n}f_{n+1}}$
$\Sigma^{\infty}_{n=2} \frac{1}{f_{n-1}f_{n}}-\frac{1}{f_{n}f_{n+1}} = \lim\limits_{N \to \infty} \Sigma^{N}_{n=2} \frac{1}{f_{n-1}f_{n}} - \frac{1}{f_{n}f_{n+1}}$
$=\lim\limits_{N \to \infty} \frac{1}{f_{1}f_{2}}- \frac{1}{f_{2}f_{3}} + \frac{1}{f_{2}f_{3}} - \frac{1}{f_{3}f_{4}}+\frac{1}{f_{3}f_{4}} -\frac{1}{f_{4}f_{5}} +...+ \frac{1}{f_{N-1}f_{N}}-\frac{1}{f_{N}f_{N+1}}$
$=\lim\limits_{N \to \infty} \frac{1}{f_{1}f_{2}} - \frac{1}{f_{N}f_{N+1}}$
$=\frac{1}{1 \times 1} - \frac{1}{\infty} = 1-0 = 1$
c)
From part (a) we know that $\frac{1}{f_{n-1}f_{n+1}} = \frac{1}{f_{n-1}f_{n}} - \frac{1}{f_{n}f_{n+1}}$
$\frac{f_{n}}{f_{n-1}f_{n+1}} = \frac{1}{f_{n-1}} - \frac{1}{f_{n+1}}$
Therefore
$\Sigma^{\infty}_{n=2} \frac{f_{n}}{f_{n-1}f_{n+1}} = \Sigma^{\infty}_{n=2} \frac{1}{f_{n-1}} - \frac{1}{f_{n+1}}$
$\Sigma^{\infty}_{n=2} \frac{1}{f_{n-1}}-\frac{1}{f_{n+1}} = \lim\limits_{N \to \infty} \Sigma^{\infty}_{n=2} \frac{1}{f_{n-1}}-\frac{1}{f_{n+1}}$
$= \lim\limits_{N \to \infty} \frac{1}{f_{1}}-\frac{1}{f_{3}} + \frac{1}{f_{2}}- \frac{1}{f_{4}}+ \frac{1}{f_{3}}- \frac{1}{f_{5}}+...+ \frac{1}{f_{N-2}} - \frac{1}{f_{N}} + \frac{1}{f_{N-1}}- \frac{1}{f_{N+1}}$
$=\lim\limits_{N \to \infty} \frac{1}{f_{1}} + \frac{1}{f_{2}} - \frac{1}{f_{N}} - \frac{1}{f_{N+1}}$
$= \frac{1}{1} + \frac{1}{1} - \frac{1}{\infty}- \frac{1}{\infty}$
$=1+1-0-0=2$
