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Computer algebra systems (CAS's) such as DERIVE, MACSYMA, MAPLE, muMATH and Mathematica can be used as powerful assistants for performing symbol manipulations in algebra and calculus. It has been suggested that these systems will benefit undergraduates in mathematics by keeping track of the details in complicated manipulations. This article shows how the symbolic calculus and graphics capabilities of the two CAS systems MAPLE and Mathematica can be used to enrich the understanding of both the theoretical and computational aspects of Taylor series. Both systems are available on Macintosh computers, and to fully utilize their capabilities requires several megabytes of RAM memory. The Mathematics Department at CSUF uses a satellite computer laboratory in our School of Natural Science and Mathematics. It currently includes 19 students work stations.
The Taylor series expansion of a function f(x) is a familiar topic of sophomore calculus. It is the foundation of much theory in analysis and applied mathematics. Unfortunately, it is often viewed by students to be difficult to understand. Using assistance from CAS to construct the coefficients of the series, draw graphs and manipulate polynomials of high degree, gives us more time to focus our attention on the underlying principles. Using MAPLE and Mathematica has generated enthusiasm for both students and teachers.
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