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) |
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} |
Given a function f (x), the Taylor series about x = a is
(x - a)^{n} |
The difference between a Taylor polynomial and a function it approximates.