A Close Look at
Taylor's Theorem
|
Taylor’s Theorem:
For each integer n and for each x, if
function f has derivatives of all order on the
open interval I (containing a):
for some value of c
between a and x. |
In simple language, Taylor’s
theorem states that the function f can be represented by n
terms of its Taylor series approximation plus a remainder term Rn(x).
Amazingly, this remainder term resembles the next term in the
series, where the (n+1)th derivative is evaluated at c (a
value between a and x) rather than at a.
The actual value of c is not that important, but its
guaranteed existence is. As
we have seen in class, it allows the possibility of finding bounds on
approximations, and proving that some functions’ Taylor Series
approximations converge to the functions on their entire domain.
Now let’s try to prove the
theorem. We will start with
a simple inequality:
As you will see, starting here
allows for a proof of the theorem where n=2.
I will then leave the rest of the proof for you as an exercise in
induction.
This initial inequality states that
the (n+1)th derivative is bounded between K and L
on the interval [a,x]. While it is unclear why this is a good starting point, it
should be clear that the remainder term depends tremendously on the
bounds of the (n+1)th derivative.
Now, we will proceed to integrate
each term of the inequality from a to x.
This yields the new inequality:
Integrating from a to x
two more times yields the following two inequalities:
Now let’s carefully examine what
we have. In the center part
of the inequality we have the difference between the function and its
second order Taylor series (centered at a).
This difference is the remainder or error resulting from using
the series approximation for the function.
The questions remain: how big is this remainder, and is there an
easy way to define it?
Let’s rewrite the inequality
simply as:
where Rn(x) is the
remainder. We already know
that K and L are the lower and upper bounds of f’’’(x)
on [a,x], hence we can write Rn(x) as:
For any given value of x, the
remainder must be a number between L and K.
By the Intermediate Value Theorem, there must exist a
value c, between a and x such that:
This completes the proof for the
case of n=2. To
generalize this proof, your task is to use the principle of induction to
show that the n case implies the (n+1) case.
