Taylor Polynomial

Theorem. Suppose that is a function for which

all exists. Let

and define

Then

Proof. Define

and . We prove that

Notice that



Thus







and

We may therefore apply l'H\opital's Rule times to obtain

Since is a polynomial of degree , its st derivative is constant; in fact, . Thus

Therefore

Definition. The polynomial is called the Taylor polynomial of degree for at .

Order of Equality

Two functions and are said to be equal up to order at if

The above theorem says that the Taylor polynomial equals up to order near .

Uniqueness

Theorem. Let and be two polynomials in , of degree , and suppose that and are equal up to order neat . Then

Proof. Let . Then is a polynomial of degree . Write

The given condition

implies that

For , we have

Thus and

Therefore

For , we have

Thus and

Continuing this way we see that

Corollary. Let be -times differentiable at , and suppose that is a polynomial in of degree , which equals up to order at . Then , the Taylor polynomial of degree for near a.

Derivation of the Taylor Expansion from Integrations by Parts

Write











Adding up these identities, we have

Taylor's Theorem

Theorem. Suppose that , are defined on and is defined by

Then

(1) there exists some in such that

(2) there exists some in such that

Moreover, if is integrable on then

(3)

Proof. Fix and . Define

for in . Then

Applying the Mean Value Theorem to on the interval there exists some in such that

Note that



Therefore

This is called the Cauchy form of remainder.

Set . Now apply the Cauchy Mean value Theorem to and there exists some in such that

Therefore

or

This is called the Lagrange form of the remainder.

If is integrable on , it follows from the Fundamental Theorem of Calculus that

Therefore

This is called the integral form of the remainder.

Some Consequences

From the equation

we have

for all

From the equation

we have