Newtons method and convergence

nobahar

Hello!
In Newton's method, I am having difficulty understanding why the convergence is quadratic.
I can imagine the closer and closer approximations moving along the x-axis as the base of a right angle triangle, but I cannot see from here how I arrive at the convergence being quadratic.
Any one know of any sources that explain this in a straightforward manner? Or would any one like to attempt to explain this for me?! It's really bugging me!
Thanks in advance.

nobahar

Should it be obvious that for, say, x² - 2, the rate of convergence of the error should be quadratic? The examples I find on the internet are far from obvious, even those that proport to be geometrical interpretations!
Any help would be really appreciated!
Thanks.