Problem

Consider the equation

and say you intend to solve it using Newton's method. If the initial value chosen is not 0, what will happen when you apply Newton's method?

Solution

We set up for a pass through using Newton's Method. We have

If the initial guess is c, the next guess will be

The next iteration will produce

and so on, so that each step produces a new approximation that is twice as far from the root (0) as before. In this way the approximations diverge, and the method fails to work.