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^{(n1)}(a) + f^{(n)}(t)dt 

 =  p_{n1}(x) + r_{n}(x) 

where
p_{n1}(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.