# What are the range of initial values using the Newton-Raphson method for #(x-9)/(x^2+1)=0# for it to converge to 9?

##### 1 Answer

As of now, what we can be sure of is that for the iteraction to converge, we must have

This is necessary, but not sufficient.

#### Explanation:

The Newton-Raphson iteration for solving

In this case, there is only one fixed point for the iteration,

For

The successive iterates

If this is smaller than 1, successive iterates will come closer and closer to 9. If larger, they will go farther and farther away.

Now

For the iteration to converge, we need

So we need

or

(since

This implies

Note that if this condition **fails**, then the point definitely moves farther and farther off from 9 on every iteration and hence it **diverges**.

This is a necessary condition for convergence, but is not a necessary one!