Page 473 - Applied Numerical Methods Using MATLAB

462 MEAN VALUE THEOREM
1
Theorem A.2. Taylor Series Theorem . If a function f(x) is continuous and
its derivatives up to order (K + 1) are also continuous on an open interval D
containing some point a, then the value of the function f(x) at any point x ∈ D
can be represented by

K k
f (a) k
f(x) = (x − a) + R K+1 (x) (A.2)
k!
k=0
where the first term of the right-hand side is called the Kth-degree Taylor poly-
nomial, and the second term called the remainder (error) term is

f (K+1) (ξ) K+1
R K+1 (x) = (x − a) for some ξ between a and x (A.3)
(K + 1)!
Moreover, if the function f(x) has continuous derivatives of all orders on D,
then the above representation becomes

∞ k
f (a)
k
f(x) = (x − a) (A.4)
k!
k=0
which is called the (infinite) Taylor series expansion of f(x) about a.