Aproximación de sucesiones definidas por recurrencia

Abstract

It is established that recurrent sequences, x_n+1 = f(x_n),which converge to a fixed point L where the derivative α = f '(L) satisfies 0 < |α| < 1 and where the rest r(x) = f(x) - L - α(x-L) satisfies r(x) = O(|x - L|^{1+ s}), for some s > 0, can be approximated as x_n ~ L + cαⁿ for a certain constant c. It is shown that such approximation might fail if only the condition 0 < |α| < 1 is assumed.