I use this approach and I think it is a good one.
Taylor polynomials are useful, can be easily motivated,
and can give a reason for a more abstract study of infinite sums.
>2. Whether or not I use the above idea I am wondering how much depth I
>should go into on tests of convergence. I teach primarily (90% or greater)
>engineering majors. I continually struggle with the fact that calculus
>curriculum has changed drastically since I was in school. I have adjusted
>to less Theorem and Proof and more real-world applications. But I also
>believe that we must proceed slowly with doing away with background
>information. We can't let technology substitute for understanding; it
>should enhance it. (Am I making any sense a all?)
I think you are right. Convergence tests will always be
important because we need to be sure of convergence
before applying any numerical method.
And if you want to test uniform convergence of
a series of functions that is not a power series
the best test is Weierstrass's test that requires
some knowledge of the convergence of numerical series.
Some proofs are a bit technical, you may skip them
(I skip them when I teach eng. majors) but you can give
a reason to justify why that rule is used.
>
>What I'm probably looking for is the answer to the question, "How important,
>in light of the technology available, to teach all the traditional
>'mathematical manipulations' associated with comparison, ratio, and root
>tests?
If you dont know whether a series is convergent it
is very risky to use technology. The harmonic series
is a good example. It seems to converge to some number
when you make calculations, even a lot of calculations.
>And how important is it to distinguish between intervals of
>convergence and radius of convergence?"
That may not be too important. Only for theoretical studies
would you be interested in what happens at the extreme points
of the interval of convergence.
I usually ask only for the radius of convergence.
Jaime
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Jaime Carvalho e Silva, Departamento de Matematica
Universidade de Coimbra, Apartado 3008, 3000 Coimbra
PORTUGAL. Phone: 351-39-7003199/50 Fax: 351-39-32568
WWW home page: http://www.mat.uc.pt/~jaimecs/index.html
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