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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
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 Rn(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.
Suppose f satisfies
for all x in some interval around a. Then,
Thus a function cannot have more than one power series in x-a that represents it.
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.
Writing out the first few terms gives
By the uniqueness theorem, this must be the Taylor series for .
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 axn where a is a constant and n is a positive integer. For example, the series for is easy to obtain as
but it is not clear at all if the following substitution
produces a useful result. This is because you would have to do a lot of work expanding powers of x2+x and collecting terms to recover a power series in x.
Even if you use a substitution of the form axn, 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
You can obtain the desired series by substitution as
but you have to be careful because this formula is not valid for all values of t. In fact this formula is only valid if . The reason for this is that the series for 1/(1+x) is only valid if and when we substitute 2t for x, the formula only makes sense if .
If a function f(x) has a Taylor series representation
then the derivative of f(x) has the Taylor series representation
obtained by differentiating each term in the series for f(x), and the series
obtained by integrating each term in the Taylor series for f(x), is an antiderivative of f(x).
William W. Farr