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The Taylor Series

Problems

Approximating Functions With Polynomials

The Remainder Term

Problem : Compute the Taylor series for f (x) = 1/(1 + x) .

The first few derivatives of the function are


f'(x) =  
f''(x) =  
f (3)(x) =  

so f (0) = 1 , f'(0) = - 1 , f''(0) = 2 , f (3)(0) = - 6 . The general case is clearly that f (n)(0) = (- 1)n n! . Hence the Taylor series for f (x) is


p (x) = x n  
  = (- 1)n x n  
  = 1 - x + x 2 - x 3 + ...  

Problem : What is the Taylor series of a polynomial p(x) = a n x n + a n-1 x n-1 + ... + a 0 ?

It is easy to check that the Taylor series of a polynomial is the polynomial itself! (All the coefficients of higher order terms are equal to 0 .)

Problem : Find the Taylor series for the function g(x) = 1/ about x = 1 .

The first couple derivatives of the function are


g'(x) = x -3/2  
g''(x) = x -5/2  
g (3)(x) = x -7/2  

so g(1) = 1 , g'(1) = - 1/2 , g''(1) = (- 1/2)(- 3/2) . We deduce that

g (n)(1) = (1)(3) ... (2n - 1) =    

Hence the Taylor series for g(x) is


x n      

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