CHAPTER 23
### Chapter 23 - More on Taylor Series

In this second
chapter on Taylor Series, we will be studying the case where the n.th derivative
of an infinitely differentiable function, does not go to zero. In such cases we
therefore have to restrict our values of x., such that for these values the
series does converge and not diverge. We will also take a look at the series
for the Exponential and Sine function.

Let
us begin with a look once again at the series for the infinitely differentiable
function, the square root of x. The theories that we are to develop will hold
true for any function raised to a positive or negative fractional exponent.

Remember we let a=1 to find the
square root of any number x. Take a close look at the fractions preceding each
term:

It seems that they are slowly going
to zero, but now look at the terms of the Taylor Series:

Since each term represents a power
of some number (x-1), then we can conclude that if this number is greater than
1, then each term in the series will get larger and larger and approach
infinity, in the infinite term.

However
if the absolute value of (x-1) were less than 1, then as the number of terms in
the series increased, the value of each term will decrease. Therefore we can
say that the following series is only valid for:

This tells us that for x in this
range of values, the infinite term will go to zero and the series will
converge. Remember, the more times you multiply a fraction by itself, the **smaller** it becomes.

Now
that we have defined the series for a certain range of numbers then we can
extend this range to include all the real numbers. For example we can find the
square root of any fraction
. This means we can find the square root of any number,
greater than 2, by taking the reciprocal of its square root. If we had to find
the
, then we just rewrite it as
. Since
then we just find the
square root of
by letting x=
, then taking the reciprocal of this value.

Now
let us move on to finding Taylor Series for the exponential and Complex Sine
Function. The Taylor series for
>can be easily found
since its n derivatives are all the same,
. The series is then:

The easiest value to choose for a
is 0 since

Since the limit as n goes to
infinity of
is zero, regardless
of what value x is, then the series is valid for any value of.

Letting x=1 and using only the
first eight terms gives us the value for e:

The calculator value for e is
2.718281828 which corresponds to an error of less than .001 % using only eight
terms. The more terms used the more accurate your answer will be.

Now
that we have found the series for
>we can find the
Taylor series for y= ln (x), which is also an infinitely differentiable
function. Fist let is find its n. derivatives:

Using the first seven derivatives
we write the following Taylor series:

Letting a equal 1:

There is one small problem here.
Though the Natural Log of x is defined for all values of x greater than zero,
the Taylor series on the other hand is only valid for 1<x<2. If x were
three for example, the series on the right would diverge as each term,
will get larger and
larger.

In
an infinitely differentiable function as in
, we assumed that the last term in:

went to zero as n
went to infinity. This was based on the fact that

However in the case of the series
for the natural log of x, the last few terms become:

Taking the limit as n goes to
infinity:

The limit of this function clearly
goes to infinity as long as the absolute value of
is greater than 1.
Conversely if
is less than 1 or
, then the last term will go to zero and the Taylor Series
will hold true. By making the modification to the Taylor Series
for ln(x), the series will converge for
. Since a =1 the series converges for:

Hence the series:

is valid for x between 0 and 2.
Despite the restricted value of x, we are still able to calculate the ln (x)
for any x, by just taking reciprocal values. If we wanted to find the ln (40),
40>2 we would have to calculate
which equals

The
Taylor Series for Sin(x) and Cos(x) are also quite easy to find. Since we know
the derivative of Sin(x) is Cos(x) and Cos(x) is -Sin(x) and we can evaluate
these functions at a=0, as Sin(0)=0 and
Cos(0)=1, the Taylor Series are as follows. First find the first few
derivatives.

Using this to write the Taylor
Series for the first eight terms is:

Letting a=1

Since the limit the infinite term
in this series goes to zero as n goes to infinity, then the series is
convergent for all values of x.

Differentiating
the series gives us the Taylor series for Cos(x):

This solution is remarkable .It
allows us to define the Sine and Cosine functions mathematically in terms of an
infinite series. You might be wondering how the series is define for any arc
length,
etc.

From the graph of the circle it is
clear that its arc length is continuous and passes throughout the same point
infinite times as it completes its rounds. For this reason our integral for the
inverse sine function could only be solved using imaginary numbers. Our study
of the Sine function began with simplicity, rose to reason, climaxed in the
abstract, fell into the imaginary, and now ends in perfection.

Our study of Taylor
Series showed us how by integrating a function
, n times, we were able to express f(x) in terms of its n derivatives
evaluated at a point a with the last terms being evaluated at some point c, between
a and x. The series took the form:

Taylor.s Theorem thus states:

The important point to realize here
is that n stands for an integer, such that a finite differentiable function can
be expressed as a series of its n derivatives evaluated at some point a. The
last term,
can be found
regardless of the value of c sine the last derivative of a finite
differentiable function is always a constant.

As
we saw, expressing finite differentiable functions in terms of a series of
powers of x turned out to be impractical. It is here where we saw the value of
extending Taylor.s theorem to infinitely differentiable functions. This would mean we would have an infinite
series with infinite derivatives. The only way to prove that such a series
could be written and would hold true for any value of x was by taking the limit
of:

If the limit was zero then we know
that
, or the last term of the series also goes to zero and therefore
the series converges for al values of x . If this limit did not go to zero then
we had to modify or restrict the values of such that the series would converge
for those values of x. We could then extend this interval for all real numbers.