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  .  

Geometric construction.

    Assume that the initial approximation   is near the root .  

Then the graph of  , intersects the x-axis at the point and the point lies on the curve near the point  .  

Define    to be the point of intersection of the x-axis and the line tangent to the curve at the point   .  

Then    should be closer to   than , as shown in the figure below.  

        

An equation relating    to    can be found if we write down two versions for the slope of the tangent line L.  

        

which is the slope of the line through    and  ,  and  

        ,  

which is the slope of the curve at the point  .

Equating these two values of the slope yields the equation  ,  which can be used to solve for  ,  and the result is  

        .  

We leave it for the reader to establish the general formula  

        .

Q. E. F.

(c) John H. Mathews 2004