CHAPTER 22
### Chapter 22 - Taylor Series

Our introductory study of Calculus ends with a short but
important study of series. The special type of series known as Taylor series,
allow us to express any mathematical function, real or complex, in terms of its
n derivatives. The Taylor series can also be called a power series as each term
is a power of x, multiplied by a different constant

are determined by the
functions derivatives. For example the constant
is based on the function.s third derivative,
on the sixth
derivative or
and so on.

It is obvious that a function with a finite number of
derivatives would have a finite number of terms as
. On the other hand ,infinitely differentiable functions such
as exponential and trigonometric functions would be expressed as an infinite
series, whose accuracy in expressing the function would be determined by the
number of terms of the series used

The proof of Taylor.s Theorem involves a combination of
the Fundamental theorem of Calculus and the Mean value theorem, where we are
integrating a function,
to get
. These two theorems say:

A quick review of the mean
value theorem tells us that:

. We therefore know:

We can now integrate the
function
once. Integrating the
left side gives us:

is just a constant,
and is equal to the n.th derivative evaluated at some point c, between a and x.

And Integrating the right
side:

Combining the two results
gives us:

If we integrate once again,
third time, we get on the left side:

On the right side we get:

Combining the two sides:

Integrating this entire mass
a fourth time, where we started with the function
gives as you may have
already guessed from the pattern:

By now the pattern should be
clear. If we integrate n successive times to eventually find f(x) we arrive at
a remarkable conclusion:

Which simplifies to:

What this is saying is that
any function can be expressed as a series of its (n-1) derivatives, each
evaluated at any point, .a., plus its n.th derivative evaluated at a point
between a and x. Since the first few terms of the series are:

One needs to not only know
the value of f(x) at f(a) but also at f.(a), f..(a), f...(a) etc... in order to
find f(x),where x, is any number other than a.

The last term of the Taylor series differs slightly from
its preceding terms. One needs to be able to calculate
for a c somewhere in
between a and x.

In a faintly differentiable function such as
the n.th derivative
is always a constant so that
is that particular
constant regardless of c in
. For example:

In this case,
=
, which will always be 3 no matter what you input in the
function. Remember n.th derivative refers to the last derivative of the
function.

In order to write or calculate a Taylor series for
we first need to
calculate its n -derivatives, which we have already done above. The Taylor
Series is defined as:

Simplifying it we get:

The easiest number to choose
for a is probably 1, though you can choose whatever number you want to for a ,
so long as its n derivatives are all defined at a.

Substituting a for 1 gives:

Now let us evaluate f(x) at
x=6 using the Taylor Series:

The sum of the series of
terms corresponds exactly; however, as you can see, writing a Taylor Series for
a faintly differentiable function is not a practical thing to do. For example
in this series we had to calculate in the last term
in order to find f(6)
or
. Now if we were able to calculate
with ease then would
it not be much easier to find
directly without
having to go through a series of calculations and summations?

The answer is yes and thus the life of finite Taylor
Series is short-lived.

The situation is a bit more complex when dealing with
infinitely differentiable functions such as
. In this case one can easily go on differentiating the
function as many times as you desire to create an infinite Taylor series where
each term is evaluated at some point a. The problem is that no matter how many
terms you use, there will always be that last term
that needs to be
evaluated at an unknown point c, between a and x. Even if we use an infinite
number of terms for our Taylor series to be evaluated at some known point a,
how de we know that our infinite plus one) or last term
does not completely
throw off our answer? From the definition of Taylor Series:

f(x) is expressed as a
series of consecutive positive integer powers of x from
. The last term is hence:

Here we have replaced the unknown
constant
with
. In the case where a=0, our last term of the Taylor series
becomes:

Taking the limit as n goes
to infinity:

The limit in the brackets is
zero as the factorial grows much faster than the exponential, remember x can
take on any value but it can never be larger than infinity or n.

No matter what value of
the last term will always be zero provided an infinite
number of terms are used in the series. The same is true of:

As a is only a constant;
however, the closer x is to a the more accurate your answer will be using fewer
terms. To conclude the Taylor Series for a function of x is:

Where the last term for a
function of finite derivatives is some constant, independent of c, multiplied
by
. In a function of infinite derivatives the last term goes to
zero after an infinite number of preceding terms. The closer x is to a then the
few the terms needed to obtain an accurate answer Therefore Taylor Series hold
true for all functions so long as one is able to evaluate its n derivatives as
a point a.

Let us move on to an important example of such infinitely
differentiable functions such as
. This is an infinitely differentiable function and as we
shall see the infinite Taylor series provides an important way of relating
square roots of a number through a series of terms than can be easily
evaluated. The answer can be obtained from the Binomial expansion, but let us
use Taylor series to show the relation between the binomial theorem and Taylor
series.

We will first use eight terms by differentiating the
function seven times:

Now writing the Taylor
Series:

Since we can evaluate each
of the n-derivatives at x=1, then we will let a=1, which simplifies the series
to:

Take note of how the
fraction preceding each term gets smaller and smaller. This is a check, that
tells us that the last terms get closer and closer to zero, and hence the
Taylor series will be defined for all x.

Let us use this series to
find the square root of 2 or

After a few simple
calculations we arrive at:

From the calculator the square root of two is:

Therefore using eight terms we found the square root of
two to an error of only .50% or half a percent. Had we used more terms the
error would have been even less. With an infinite amount of terms the error
would be zero or next to nothing.