The famous Fibonacci numbers is a sequence defined by a0 = a1 = 1, ab =an-1 + an-2 for n ≥ 2. Compare this sequence with the coefficients of the power series centered at 0 of the function \(\displaystyle \frac{1}{z^2+z-1}\).
\(\displaystyle \frac{1}{z^2+z-1}\) = \(\displaystyle \frac{1}{-1 + z(z+1)}\) = \(\displaystyle \frac{-1}{1 - z(z+1)}\)
\(\displaystyle \frac{a}{1 - x}\) = ∑ xn
\(\displaystyle \frac{-1}{1 - z(z+1)}\) = - ∑ (z(z+1))n
\(\displaystyle \frac{1}{z^2+z-1}\) = \(\displaystyle \frac{1}{-1 + z(z+1)}\) = \(\displaystyle \frac{-1}{1 - z(z+1)}\)
\(\displaystyle \frac{a}{1 - x}\) = ∑ xn
\(\displaystyle \frac{-1}{1 - z(z+1)}\) = - ∑ (z(z+1))n