**Theorem (Taylor's Formula with Remainder):**

Suppose that f is a real function on the closed interval [c, d] and n is a positive integer.

Furthermore, suppose that the n+1st derivative of f is everywhere finite on the open

interval (c, d) and that the nth derivative of f is continuous on the closed interval [c, d].

Let a be any point in the interval [c, d]. Then for every point x in [c, d] with x not equal to a

there exists a point

which is strictly between x and a, such that the following formula holds:

=

where the error is given by

=

**Definition****:**

A real-valued function f defined on an interval [c, d] is said to belong to

on [c, d]

provided that f has derivatives of all orders defined at every point of [c, d].

**Note:**

This means that all of the following derivatives exist on [c, d]:

,

,

,...

**Definition (of a Taylor's Series)****:**

If f belongs to

on [c, d]

and we pick a point a such that c < a < d, then the **Taylor's**

**series**** about a generated by f** is the following power series:

**Note:**

If a = 0, the Taylor's series is called a Maclaurin series. It simplifies to:

**Note:** It must always be asked whether a given Taylor's series converges and whether it

converges to f(x). The following theorem helps to answer this.

**Corollary (Taylor's Series Convergence):**

If f belongs to

on [c, d]

and we pick a point a such that c < a < d, then the Taylor's

series converges to f(x) if and only if

=

(where the error (remainder) is defined as shown above)

**Note:** A typical way to show that this limit is zero is to show that the derivatives of f are bounded,

at least in some interval about a.

**Note:** If we know that a Taylor's series converges to f(x) then we have that:

=

=

If a = 0, this simplifies to:

=

=

**Note:** Since we cannot do an infinite sum on the computer, we typically compute a partial sum

and use Taylor's formula with Remainder to try to produce a bound on how bad the error can be.

The actual error, of course, might be much less that our bound. The bound provides a worst case

for the error.