Przemyslaw Bogacki

Old Dominion University

Department of Mathematics and Statistics

Norfolk, VA 23529

bogacki @ math.odu.edu

In 1992 the author, along with two colleagues, began developing a computer-based calculus course sequence, under a grant from the State Council for Higher Education in Virginia [1,2]. To enhance our students' learning of the infinite series material, a computer laboratory activity devoted to the subject was created.

In the Summer 1994, the author developed computer activities intended to provide an intuitive interpretation to some of the fundamental notions involved in studying infinite series and Taylor polynomials. These activities are described in the following sections.

When Mathcad is in "automatic mode", changes made by a user go
into effect immediately, affecting the rest of the document. By
typing a different expression for *a_n*, different
categories of series can then be illustrated: p-series,
geometric series, alternating series etc.. The values assigned
to the range variables *n,i* and *m* can also be
modified to control which terms and partial sums are calculated,
plotted and tabulated, respectively (subject to the restriction
that the sets of values taken by *i* and *m* both be
subsets of the set of values of *n*.) The instructor can
perform such modifications to demonstrate the behavior of
different series, and encourage students to experiment with
different series.

The "arrow" plots of *a_n* and *S_n* are designed to
show students how the terms of the series "build up", forming
partial sums. For alternating series, these plots will aid a
student's understanding of cancellations that take place when
the partial sums are formed. Since students tend to have
difficulty in distinguishing between the sequence of terms, and
that of partial sums, we find illustrations that clearly
contrast the two concepts (while emphasizing the relationship
between them) very useful.

Another crucial distinction that has to be stressed is that
between the *convergence* of the sequence of *terms*,
and *convergence* of the *series*. The arrow plots and
tables can be very helpful to provide a valuable visual and
numerical illustration. In particular, the professor can
demonstrate divergence of a series whose sequence of terms
converges to zero (which tends to be hard to grasp for many
students).

Upon loading the Mathcad document (see Fig.
2) the instructor should briefly explain the items present
on the screen: the function definition, its plot, and a "table"
of *f*(0) and the first four derivatives of *f* at 0.
Also it should be noted that *P*(*x*)=*a*_0, a
polynomial of zeroth degree has been defined, and its value,
and those of its first four derivatives, at *x*=0 are
tabulated alongside those of *f*(*x*). The students
should be asked a question: **For what value of the constant
***a*_0** will the polynomial
***P*(*x*)=*a*_0** "mimic" the behavior of
***f*(*x*)** close to the value ***x*=0?
Students are likely to produce the answer:
*a*_0=*f*(0)~6.306. This is the best we can do with a
polynomial of degree 0. We now raise the degree of the
polynomial to 1 setting the new coefficient *a*_1 initially
to **zero**. Question: **How should we use this extra
freedom to do a better job following ***f*(*x*)**
around ***x*=0** with our ***P*(*x*)? After
some experimentation, students will answer that *a*_1=*f
'*(0)~6.814 . We then remark that we now have a match in the
first two rows of the derivative table (see Fig.
3).

We can get a match in more rows if we raise the degree again.
Set *a*_2=0 and add the *a*_2 *x*^2 term to the
expression defining *P*(*x*). Again, let students
experiment with the value of *a*_2. They will quickly
discover that . The instructor may want to take a break from
the activity, and on the board show that for any polynomial

we have

. The activity will be concluded by raising the degree twice more, and setting the appropriate coefficient values. The final screen is presented in Fig. 4.

We believe this activity very well exemplifies Mathcad's potential in the classroom. Mathcad has a unique ability to arrange objects like assignments, expression evaluations and graphs in a logical (rather than interface-enforced) fashion. It was this feature, combined with Mathcad's dynamic response to user's input, that allowed this activity to occupy just one screen (without scrolling), while revealing a lot about the nature of Taylor polynomial approximation (in both numerical and visual fashion).

Fig. 1. The Mathcad screen corresponding to the Infinite Series
activity.

Fig. 2. The initial screen from the Taylor Polynomials
activity.

Fig. 3. An intermediate screen from the Taylor Polynomials
activity.

Fig. 4. The final screen from the Taylor Polynomials activity.