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The Taylor Series

The Remainder Term

Problems

The Remainder Term, page 2

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We now use integration by parts to determine just how good of an approximation is given by the Taylor polynomial of degree n , p n(x) . By the fundamental theorem of calculus,

f (b) - f (a) = f'(t)dt    

Integrating by parts, choosing - (b - t) as the antiderivative of 1 , we have


f'(t)dt = - f'(t)(b - t)|a b + f (2)(t)(b - t)dt  
  = f'(a)(b - a) + f (2)(t)(b - t)dt  

Again, integrating by parts yields


f (2)(t)(b - t)dt = - f (2)(t) + f (3)(t) dt  
  = f (2)(a) + f (3)(t) dt  

Putting these equations together, we have

f (b) = f (a) + f'(a)(b - a) + f (2)(a) + f (3)(t) dt    

Continuing the process, we arrive at

f (b) = f (a) + f'(a)(b - a) + ... + f (n-1)(a) + f (n)(t) dt    

Substituting x for b , we have an expression for f (x) , called Taylor's formula at x = a , involving the familiar Taylor polynomial of degree n - 1 for f and an integral called the remainder term and denoted by r n(x) :

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