Past exam of the year 1 of the computer science course of the University of Oxford / 2016 / Discrete mathematics, continuous mathematics and probability (resit) / 6 / f / Solution

For $1<x_0<2$, the same Newton's method formula givesThus the first step overshoots the root to $x_1>2$. Part (e) then applies from index $1$: the remaining sequence decreases to $2$. The first step can be very large when $x_0$ is close to $1$, where the derivative vanishes, but every starting value strictly between $1$ and $2$ still gives convergence to $2$. The entire sequence is not decreasing, since $x_1>2>x_0$.