Abstract
This paper is an attempt to illustrate and emphasize three points. First, that modern technology allows one to encourage students to ask and answer questions heretofore impossible to address fruitfully. Second, many (probably most) classical methods are as necessary as ever, but perhaps in a different setting, and third, simple calculus and a bit of reflection is amazingly effective. Although the examples presented are specific, I have attempted to approach each in a manner that has general applicability. Throughout I have tried to show how modern technology can provide insight and foster a spirit of inquiry. Machines do some things very well, some poorly. The same is true of humans. The two are often complementary; it seems best to search for uses of each that capitalize on their individual strengths. The topic of infinite series provides convenient examples. Given an infinite series, one asks two things. Does it converge, and if so, to what? The former question has been the concern of most calculus texts, while the latter has usually been left in abeyance. This is unfortunate for the applications-oriented student, for it is the value of the series that is most often the thing of primary concern.
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