cp /math/calclab/MA1023/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 another power series you have seen in class.

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

Another example is the power series for (or ), which is

The exercises also use the series for and , which you haven't seen yet, but will very soon.

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 .

- For the
following functions, compare the partial sums of power series with
base points up to
various orders obtained directly with those obtained by applying
addition, subtraction, substitution, multiplication, or division to
the power series for , , , or the geometric
series for . See the examples in the
`Getting Started`worksheet.- .
- .

- Use substitution followed by integration to generate the first
three terms in the power series with base point for
. Start with the series for . Do you get the same
result if you start with the series for and use integration
followed by substitution?
- You know that the derivative of is . Is it
true that if you differentiate the power series with base point
for you
get the power series for ? Investigate this by
differentiating the power series for
and comparing it to the power series for . Use at least
five terms in your comparison.

2005-04-01