Taylor Series

Let be a function defined at and in some neighborhood of . Specifically suppose there is an interval
    (, constant)
in which is continuous and has derivatives of all orders. Let
Then the power series (in powers of )
    (1)
with coefficients
    (2)
is called the Taylor series for the function about . [To apply equation (2) in the case , we use the definitions
and ; in other words ]
When the Taylor series for about actually converges to the value of , that is, when
  (x in I)   (3)
is said to be analytic at .

The overwhelming majority of functions (e.g. algebraic, exponential, trigonometric, hyperbolic) are analytic: They can be represented by their
Taylor series or approximated by partial sums of the series. Furthermore, in some cases (in particular, for exponential, trigonometric,
and hyperbolic functions), the interval of convergence, I, on which (3) holds, is , the entire real line.

The case is of special interest. If is analytic at , eqn. (3) takes the form
    (4)
This power series is called the Maclaurin series for (Recall that )

Example 1

Find the MacLaurin series for .

SOLUTION:


Further derivatives repeat this pattern. Substitution in eqn. (4) then gives



  (5)
It may be shown (e.g., see references [1,5,6]) that eqn. (5) is true for all values of x.

Example 2

Find the Taylor series for about .
SOLUTION: Since can be any (fixed) number, the values of ,
        ,
        ,
        ,
        ,...
cannot be simplified. Thus, for an arbitrarily given number , the Taylor series here has the form

REMARKS
  1. Again, the preceding series may be shown to converge for all values of .
  2. When , this series reduces to the Maclaurin series in eqn. (5).

Example 3

Find the Maclaurin series for where and is an arbitrary fixed real number.

SOLUTION:The derivatives of are

For we have
.

We can now write eqn. (4) for the function ; but first we point out that the Maclaurin series in this example converges
(to ) only on a finite interval, namely, Thus we have

    (6)

The series in eqn. (6) is called the Binomial series. We note two special cases of eqn. (6) which occur frequently in applications:
, a positive integer:
Let us rewrite eqn. (6) as
    (7)

Now is a fixed positive integer. In the term of the series the last factor of the numerator is .
The next term , will have an additional factor, , and so will all subsequent terms have this factor. In other words, the coefficients of the powers , , etc. will all be zero. Thus the series has only a finite number of terms, reducing to a polynomial of order m:
    (8)
where
    (k=1,2,..,m)
are called the binomial coefficients. Eqn. (8) holds for all because it is an algebraic identity. (Try it for yourself: expand etc.). Eqn. (8) is called the binomial theorem.

:
Setting in (6) gives
    (|x|<1)

Example 4

Find the Maclaurin series for
SOLUTION: To proceed straightforwardly we first get
Then will be a sum of four terms, each a product; will involve still more terms; etc. This example shows that it is not always easy to calculate the Taylor coefficients for a given function . What to do?

It is tempting to multiply the power series like polynomials, that is, assume that the series for the product of two functions is equal to the product of the series, then carry out term-by-term multiplication of the two series. This would enable us to use the series expansions for and , already found in Examples 1 and 3.
In fact this procedure has been proved to be mathematically sound. Similarly, other operations on power series (such as term-by-term differentiation and integration, addition of two series, and division of two series) can also be carried out under suitable conditions.

Returning to the function given here, we use eqns. (6) and (5) to write


                     
          

REMARKS
  1. The result holds for (that is, for ) because the general mathematical justification for the manipulation includes the condition that the series representation for the product is valid on the common interval of convergence of the two simpler series. (Recall eqn. (5) holds for and eqn. (6) holds for ).
  2. For the term-by-term multiplication above we have retained only enough terms to write the product series up through the term in . In principle we can find as many terms in the product series as we wish. Similarly, one could compute as many derivatives of to employ the straightforward Taylor series method. An important difference, however, is that in the manipulative method carried out here the further calculations needed are only algebraic, while the straightforward Taylor series method requires increasingly complicated differentiation. In any case it is helpful to be aware of alternative methodologies.

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