Taylor Theorem

                                                        Taylor Theorem

Proof:
     We assume that f satisfies the hypotheses of the theorem and define the Taylor Polynomial about a of degree n :
 
This polynomial and its first n derivatives match the function f and its first n derivatives at x = a. We do not disturb that   matching by adding another term of the form   
 where K is any constant, because such a function and its first n derivatives are all equal to zero at x = a.
Therefore, the new function
                                                

and its first n derivatives still agree with f and its first n derivatives at x = a.
   We now choose that particular value of K that makes the curve
  agree with the original curve Y = f(x) at x =b        . 
This can be done: we need only satisfy
                                                
or
                                                                   
, so that  With K defined by Eq. above, let 
F(x) measures the difference between f and approximating , for each x in [a, b], or in [b, a] if b<a. To simplify  function 
the notation, we assume a<b, so that a is the left endpoint of all intervals mentioned. The same proof is valid if a is the right endpoint, instead of the left endpoint  for example:
 
The remainder of the proof makes repeated use of Rolle s
 are  Theorem. First, because F(a) = F(b) = 0 and both
continuous on [a, b], we know that
 in (a, b). for some 
 are  and both  Next, because
we know that  continuous on

 in  for some 
implies Rolle s Theorem, applied successively to
the existence of 
   such that  in  
   such that  in  
                                                                                         :
   :
                                                                            :
 such that  in  
 and differentiable  is continuous on   Finally, because
 Rolle s Theorem implies  and    on
 such that in   That there is a number

 Where we differentiable

 , for n + 1 times  and  So,
We get
 
Equations,
  and 
Together lead to the result

  for some number  

Combining Equations,
     
And
 
 for some number   
 We have,
 
Or

 for some c between a and b.

Notation:
   If f has derivative of all orders in an open interval I containing a, then for each positive integer n and for each x in I,
  where
 for some c between a and x. 
   The corollary follows at once from Taylor s Theorem because the existence of derivatives of all orders in an interval I implies the continuity of those derivatives and we have merely replaced b by x in the final formula.

 is called the remainder of order n : it s      The function 
is the Taylor   where  the difference 
Polynomial of degree n used to approximate f (x) near x = a.  This difference, also called the error in the approximation
can often be estimated by using Equation

for some c between a and x. 

, for all x in some interval around   When

x = a, we say that the Taylor-series expansion for f (x) converges

to f (x) on that interval and write