Exercises in Iterational Asymptotics II

TL;DR

This paper investigates the asymptotic behavior of nonlinear recurrences, including oscillatory and monotonic convergence, providing detailed analysis and constants for specific functions.

Contribution

It offers new detailed asymptotic analyses of nonlinear recurrences, extending previous work and calculating constants for convergence behaviors.

Findings

Asymptotic formulas for oscillatory convergence to fixed points.

Calculation of convergence constants for specific nonlinear functions.

Analysis of monotonic convergence in complex recurrences.

Abstract

The nonlinear recurrences we consider here include the functions $3x(1-x)$ and $\cos(x)$, which possess attractive fixed points $2/3$ and $0.739...$ (Dottie's number). Detailed asymptotics for oscillatory convergence are found, starting with a 1960 paper by Wolfgang Thron. Another function, $x/(1+x\ln(1+x))$, gives rise to a sequence with monotonic convergence to $0$ but requires substantial work to calculate its associated constant $C$.