Exercises in Iterational Asymptotics II

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