MathDB
Prove that there exists an integer k [ILL 1974]

Source:

January 2, 2011
trigonometrylimitalgebra proposedalgebra

Problem Statement

Let f:R→Rf : \mathbb R \to \mathbb R be of the form f(x)=x+ϵsin⁡x,f(x) = x + \epsilon \sin x, where 0<∣ϵ∣≤1.0 < |\epsilon| \leq 1. Define for any x∈R,x \in \mathbb R, xn=f o … o f⏟n times(x).x_n=\underbrace{f \ o \ \ldots \ o \ f}_{n \text{ times}} (x). Show that for every x∈Rx \in \mathbb R there exists an integer kk such that lim⁡n→∞xn=kπ.\lim_{n\to \infty } x_n = k\pi.
Back to ProblemsView on AoPS