The curve xn+1 = f (xn) intersects the curve xn+1 = xn at x0. The expansion of

Question:

The curve xn+1 = f (xn) intersects the curve xn+1 = xn at x0. The expansion of xn+1 about x0 is xn+1 – x0 = β (xn – x0) where β = (df/dx) at x = x0.
(a) Describe the geometrical sequence that the successive values of xn+1 – x 0 forms.
(b) Show that the intersection is stable when |β| < 1 and unstable when |β| > 1.