Theorem (Newton-Raphson Theorem).  Assume that and there exists a number , where .  If   , then there exists a such that the sequence defined by the iteration  

        for      

will converge to for any initial approximation  .  

Derivation.

Newton's method is based on linear approximation or the Taylor polynomial of degree expanded about  .  

Which means that we have the equation

For this Taylor polynomial, the Lagrange form for the remainder is  ,  and we write  

(1)    .  

We seek the value    which makes  .  

Substituting    in equation (1) results in

(2)    .  

Since the remainder term is for theoretical consideration and not for numerical computation, we must chop it off or truncate the right side of equation (2).  But then it is not an equation, only an approximation.

(3)    .  

If we wish to "solve for p" it is impossible.  But if we change the to and change to , then we will have an equation to solve.

(4)    .  

Can Mathematica be used to solve for ?  Mathematica's represented the error term for Taylor polynomial as  .  

We can eliminate with the command .

We can substitute for with the command .  

Now we have equation (4) which we can solve for the next iteration  .  

It would look nicer if we dig out the formula in the two braces.

Expand this solution in its familiar form.

So the next iteration is  .   

Q. E. D.

(c) John H. Mathews 2004