Chapter 9.10, Problem 92E

Using Fibonacci Numbers Show that the Maclaurin series for the function

$g(x)=\frac{x}{1-x-x^2}$ is

${\displaystyle \underset{n=1}{\overset{\infty }{\sum }}{F}_n{x}^n}$ where F_n is the nth Fibonacci number with $F_1=F_2=1$ and $F_n=F_{n-2}+F_{n-1}$, for $n\ge 3$. (Hint: Write

$\frac{1}{1-x-x^2}=a_0+a_{1}x+a_2{x}^2+\cdots$ and multiply each side of this equation by $1-x-x^2$.)