Problem 33

Given information

According to the question with help of Newton's method for approximating a root  of the equation  , and from an initial approximation  we can obtain successive approximations  , where

 

Step-by-step explanation

 

With and , it takes the form:

 

Now we need to find , it is defined as

Next step is to write the first order polynomial at

 

now becomes:

 

Substitute and

 

The problem states that is a root of meaning :

 

Now transforming Newton's equation, we achieve:

 

We plug this in :

 

 

Take its absolute value :

 

Finally we know that so let's take to be the smallest value it can be:

 

Final form of is:

 

Plug it in Taylor's Inequality: