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