For an times differentiable function and for and in an open
interval on the real line, there exists between and we have
where the remainder is given by
Applications of Taylor's theorem
The practical application of Taylor's theorem is to provide a ready alternate representation of a function by expanding that function about a given point. The
number of terms of the Taylor series expansion reflects the number of continuous derivatives that the function being expanded has at the point that it is being
Taylor's theorem forms the foundation of a number of numerical computation schemes, including the approximation of smooth functions, the formulation of finite-difference
methods, and the formulation of optimization algorithms.
The complex form of the Taylor's series for a complex valued function of a complex valued function converges if and only if the function is analytic within a neighborhood of the point about, the function is being expanded.
Taylor series as an infinite series in 1D
A real valued function can be expressed in terms of the value of the function and its derivatives at any point . In one variable this is
where ! denotes factorial (e.g., ).
By the ratio test of convergence, the Taylor series converges for values of where the ratio of the -th and the -th terms is less than 1. That is:
The Maclaurin series is the special case where b=0.
Proof of the 1D Taylor Theorem
For we consider the
We define a new function such that .
Differentiating with respect to we obtain
Now, we construct the function such that and
Because is not constant, by Rolle's theorem must have an extremum at
some value between and , so and
solving for the remainder , yielding
Arbitary N Case
This same method generalizes to the case of arbitrary ,
We define a new function such that as
As above, we differentiate with respect to to yield
As in the case, we construct the function such that and
Because is not constant, it must have an extremum at a point somewhere
Solving for completes the proof
Proof of the infinite series form of Taylor's theorem in 1D
By the fundamental theorem of calculus we note that
Our method of constructing the Taylor's series will be by repetitive integration by parts of the remainder term, where we
We integrate the integral remainder term by parts, integrating the term and differentiating
the term we have
This process may be applied repetitively to yield the familiar form of the Taylor expansion. Suppose this has been done
times, then the remainder term
Taylor's series of an analytic function
Given a function analytic inside some region of the complex plane,
then we may write
where the series converges in a disc, centered at the value , contained within .