cp ~bfarr/Powseries_start.mws ~

You can copy the worksheet now, but you should read through the lab
before you load it into Maple. Once you have read to the exercises,
start up Maple, load
the worksheet `Powseries_start.mws`, and go through it
carefully. Then you can start working on the exercises.

The number is called the base point of the power series. In this lab, we will consider only the special case . Historically, power series have been used most often to approximate functions that do not have simple formulas. One example is the exponential function, whose power series is given below along with some other important power series you have seen.

The most familiar example of a power series is the geometric series.

Once you have a convergent series representation for a function, it can be manipulated in several ways to generate convergent power series representations of related functions. The rest of the Background describes several different techniques for generating power series representations of functions that are related to power series that are already known. The four techniques are substitution, multiplication and division, integration, and differentiation.

Writing out the first few terms gives

If you use such a substitution, you have to be
careful if the series is only valid for a finite interval. For
example, suppose you wanted to find the power series for

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 . In fact this formula is only valid if . The reason for this is that the series for is only valid if and when we substitute for , the formula only makes sense if .

so we have the following power series representation.

Note that the interval of convergence is exactly the same as for the original series, neglecting the behavior at the endpoints.

Integrating a power series term-by-term is very similar, but you may
have to include a constant of integration. For example, integrating
the power series representation for term by term gives

You would have to set to make the right hand side the power series representation for .

- Verify that substitution and multiplication work as described
above to generate partial sums of power series (with base point a=0) for the
following functions. That is,
compare the partial sums of power series up to various orders
obtained directly
with those obtained by substitution, multiplication, or division.
- .
- .

- Use substitution followed by integration to find the first four non-zero terms in the power series representation for . Start with the series for .
- In the book it says that power series can also be added term by
term. Try the following two Maple commands and verify that the
output is the same.
> 1/2*Taylor(exp(x),x=0,5)+1/2*Taylor(exp(-x),x=0,5); > Taylor((exp(x)+exp(-x))/2,x=0,5);

Can you explain why the power series only contains even powers of ?

2003-01-28