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?
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.
The Newton-Raphson iteration for solving
In this case, there is only one fixed point for the iteration,
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.
For the iteration to converge, we need
So we need
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!