Order of convergence of fixed-point to Newton's method

Hi,

In a problem I first show that the order of convergence of simple iteration is 1 and that in order for it to converge I need for all .

From this I must show that Newton's method has an order of convergence of 2. I usually show this by starting with a Taylor expansion etc.

I'm thinking that I could perhaps start by saying that fixed-point iteration will converge if for all and since Newton's method has that ,

I get,

Since Newton's method is defined as

,

we get that,

Then,

as but that doesn't show that Newton's is a second order method..

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I've also started by saying that since simple iteration is of order 1, we have that,

where and ,

but it does not lead me to anything good.

A nice hint would be great! Thanks.