How do we know the Newton-Raphson method produces a better approximation than the initial point.

Question

I know this method works by finding the root of the tangent line to the initial point, which is the best linear approximation of the function at that point. And I also know the closer the initial point to the root, the better the approximation, since a tangent line approximation is more accurate for nearby values.

But how do we know the root of the tangent line always produces a value that is closer the the root than the initial point? I haven't seen any author provide an explanation on this other than providing some diagrams. Is it possible that sometimes it gives an value that is slightly further away from the initial point (provided the initial guess is fair)?

Answer 1

The short answer is that it doesn't. You don't know, because it depends on the function and the initial guess.

Refer to the following question and answer: Examples of when Newton's Method will fail? In that answer, you can see that for certain functions, you could have nonconvergent behavior, even repelling fixed points.