Taylor's theorem
In
calculus, Taylor's theorem, named after the
mathematician Brook Taylor, who stated it in 1712, gives the approximation of a
differentiable function near a point by a
polynomial whose coefficients depend only on the derivatives of the function at that point. This result was first discovered 41 years earlier in 1671 by James Gregory.
Encyclopedia
In
calculus,
Taylor's theorem, named after the
mathematician Brook Taylor, who stated it in 1712, gives the approximation of a
differentiable function near a point by a
polynomial whose coefficients depend only on the derivatives of the function at that point. This result was first discovered 41 years earlier in 1671 by James Gregory.
Taylor's theorem in one variable
The most basic example of Taylor's theorem is the approximation of the
exponential function near
x = 0. Namely,
The precise statement of the theorem is as follows: If
n ≥ 0 is an integer and
f is a function which is
n times continuously differentiable on the closed interval [
a,
x] and
n + 1 times differentiable on the open interval , then we have
Here,
n! denotes the
factorial of
n, and
Rn is a remainder term which depends on
x and is small
if
x is close enough to
a. Several expressions for
Rn are available.
The
Lagrange form of the remainder term states that there exists a number ξ between
a and
x such that
This exposes Taylor's theorem as a generalization of the
mean value theorem. In fact, the mean value theorem is used to prove Taylor's theorem with the Lagrange remainder term.
The Cauchy form of the remainder term is
This shows the theorem to be a generalization of the
fundamental theorem of calculus.
For some functions
f, one can show that the remainder term
Rn approaches zero as
n approaches ∞; those functions can be expressed as a
Taylor series in a neighbourhood of the point
a and are called analytic.
Taylor's theorem is also valid if the function
f has
complex values or vector values. Furthermore, there is a version of Taylor's theorem for functions in several variables.
For complex functions analytic in a region containing a circle
C surrounding
a and its interior, we have a contour integral expression for the remainder
valid inside of
C.
Taylor's theorem for several variables
Using multi-index notation , Taylor's theorem can be generalized to several variables as follows. Let
B be a
ball in
RN centered at a point
a, and
f be a real-valued function defined on the closure having
n+1 continuous partial derivatives at every point. Taylor's theorem asserts that for any ,
where the summation extends over multi-indices α.
The remainder terms satisfy the inequality
for all α with |α|=
n+1. As was the case with one variable, the remainder terms can be described explicitly. See the proof for details.
Proof: Taylor's theorem in one variable
We first prove Taylor's theorem with the integral remainder term.
The
fundamental theorem of calculus states that
This proves the theorem for
n = 0.
Integration by parts yields the case
n = 1:
By repeating this process, we may derive Taylor's theorem for higher values of
n.
This can be formalized by applying the technique of
induction. So, suppose that Taylor's theorem holds for a particular
n, that is, suppose that
We can again rewrite the integral using integration by parts. An antiderivative of
n as a function of
t is given by −
n+1 / , so
Substituting this in proves Taylor's theorem for
n + 1, and hence for all nonnegative integers
n.
The remainder term in the Lagrange form can be derived by the
mean value theorem in the following way:
The last integral can be solved immediately, which leads to
Proof: several variables
Let
x= lie in the ball
B with center
a. Parametrize the line segment between
a and
x by
u=
a+t. We apply the one-variable version of Taylor's theorem to the function
f:
By the chain rule for several variables,
where is the multinomial coefficient for the multi-index α. Since , we get
The remainder term is given by
The terms of this summation are explicit forms for the
Rα in the statement of the theorem. These are easily seen to satisfy the required estimate.
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