Chapter 11.P, Problem 24P

(a) Show that the Maclaurin series of the function

$f\left(x\right)=\frac{x}{1-x-x^2}$ is ${\displaystyle \underset{n=1}{\overset{\infty }{\sum }}{f}_n}{x}^n$

where $f_n$, is the nth Fibonacci number, that is, $f_1=1$, $f_2=1$, and $f_n=f_{n-1}+f_{n-2}$ for $n\ge 3$.

[Hint: Write $x/\left(1-x-x^2\right)=c_0+c_{1}x+c_2{x}^2+\mathrm{...}$ and multiply both sides of this equation by $1-x-x^2$.]

(b) By writing $f\left(x\right)$ as a sum of partial fractions and thereby obtaining the Maclaurin series in a different way, find an explicit formula for the nth Fibonacci number.