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.