Chapter 11, Problem 22P

(a) Show that the Maclaurin series of the function

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

wheref_n is the nth Fibonacci number, that is, f₁ = 1, f₂ = 1, and f_n = f_n−1 + f_n−2 for n ≥ 3. [Hint: Write x/(1 – x − x²) = c₀ + c₁x + c₂x² + ⋯ and multiply both sides of this equation by 1 – x − x².]

(b) By writing f(x) 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.