Hi, I have the following exercise... I have solved the first question, but I don't know about the second question. Thanks for whatever help you have give me!

Since early electronic computers had no automatic division, it was necessary to accomplish this by a process of calculating reciprocals using only addition, subtraction and multiplication.

1) Show that the Newton-Raphson method iteration function for $\displaystyle r(x)=x^{-1}-R$, where $\displaystyle R>0$, is

$\displaystyle F(x)=x(2-Rx)$.

Verify that $\displaystyle x*=R^{-1}$ is a fixed point of $\displaystyle F(x)$.


And here is the question that I have no clue about (I don't need any help with the question above):

"2) Show further that the iteration satisfies

$\displaystyle 1-Rx_{n+1}=(1-Rx_n)^2$,

and deduce that
$\displaystyle 1-Rx_{n+1}=(1-Rx_0)^{2^{n+1}}$.

Hence show that $\displaystyle Rx_n -> 1$ as n approaches infinity if and only if $\displaystyle 0<Rx_0<2$.
Deduce that $\displaystyle x_n->R^{-1}$ as $\displaystyle n$ approaches infinity if and only if $\displaystyle 0<x_0<2/R$.

Note: you might need to illustrate this conclusion graphically."