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The general form of the Taylor series representation with base point
*x*=*a* of a function *f*(*x*) is given by the following

**Definition 1**

Suppose that *f*(*x*) is a smooth function in some open interval
containing *x*=*a* and that for every fixed value of *x* in this
interval, it can be shown that the Taylor polynomial remainder
*R*_{n}(*x*) goes to zero as . Then the Taylor
series representation with base point *x*=*a* of the function *f*(*x*) is
given by

In class we will derive the following four important Taylor series.

where the last series is really the formula for the sum of a geometric series in disguise.

Once you have a Taylor series representation for a function, it can be used in several ways to generate Taylor series representations of related functions. This is because of the following theorem.

**Theorem 1**

Suppose *f* satisfies

The rest of the Background describes several different techniques for generating Taylor series of functions that are related to Taylor series that we already know. The four techniques are substitution, multiplication and division, integration, and differentiation. We have already seen examples of integration and differentiation with Taylor polynomials, but we haven't talked about the first two techniques yet.

The most commonly useful Taylor series have base point *x*=0 and that
is what we will focus on in this lab. The technique of substitution is
most useful if the substitution is of the form *ax*^{n} where *a* is a
constant and *n* is a positive integer. For example, the series for
is easy to obtain as

Even if you use a substitution of the form *ax*^{n}, you have to be
careful if the series is only valid for a finite interval about the
base point. For example, suppose you wanted to find the Taylor series
with base point *t*=0 for the function

**Theorem 2**

If a function *f*(*x*) has a Taylor series representation

- 1.
- Verify that substitution and multiplication work as described
above to generate Taylor series (with base point a=0) for the
following functions. That is,
compare the Taylor polynomials for various orders obtained directly
with those obtained by substitution, multiplication, or division.
- (a)
- .
- (b)
- .
- (c)
- .
- (d)
*f*(*x*) =*x*/(1+*x*).- (e)
- .

- 2.
- Find the first five terms of the Taylor series representation
with base point
*x*=0 for the function in two ways. First, by computing the Taylor polynomial directly and, second, by using the series for 1/(1+*x*). What is the interval of convergence of the series representation? You might want to plot some Taylor polynomials to check your answer for the interval of convergence. - 3.
- Use substitution followed by integration to generate the first
four terms in the Taylor series with base point
*x*=0 for . (Hint - start with the series for 1/(1+*x*).) - 4.
- Can you find the sum of the series if ?
- 5.
- In the background section we only considered multiplication of
series by polynomials. Suppose you wanted to generate the Taylor
polynomial of order ten with base point
*x*=0 for the function . Can you do this by multiplying Taylor polynomials for and ?

11/29/1998