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The Taylor Series

Terms

Introduction and Summary

Approximating Functions With Polynomials

Taylor's Formula  -  A formula expressing a function in terms of a polynomial approximation and an error (remainder) term. Explicitly, given a function f (x) and a real number a ,


f (x) = f (a) + f'(a)(x - a) + ... + f (n-1)(a) + f (n)(t) dt  
  = p n-1(x) + r n(x)  

where p n-1(x) is a Taylor polynomial and r n(x) is the remainder term
Taylor Polynomial  -  The approximation of a function f (x) around a point x = a by a polynomial

p n(x) = f (a) + f'(a)(x - a) + f''(a)(x - a)2 + ... + f ( n)(a)(x - a)n    

for some n≥ 0 .
Taylor Series  -  Given a function f (x) , the Taylor series about x = a is

(x - a)n    

The Taylor polynomials for f about a are the partial sums of this series.
Remainder Term  -  The difference between a Taylor polynomial and a function it approximates.

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