Chapter 8: Infinite Sequences and Series
Section 8.1 - Sequences
Section 8.2 - Series
Section 8.3 - Convergence Tests
Section 8.4 - Power Series
Section 8.5 - Taylor Series
Chapter 8 makes use of the limit concept and even the integral, but these uses that seem to be the connection to calculus are not the real reason for including a chapter on infinite sequences and series in the course. There is a great deal of history and philosophy lurking in the topics of this chapter. Interested students might find the text The History of the Calculus and its Conceptual Development by Carl B. Boyer an interesting read at this point in their intellectual growth.
The notion of the limit of an infinite sequence merely paves the way for a theory of infinite series. And history shows that there was a need for such a theory, and this history goes back to the ancient Greeks. The Pythagoreans were a philosophical, not a mathematical, school, which believed that all things in the universe could be identified and explained via the counting numbers. But it didn't take long to discover that in an isosceles right triangle with legs 1 unit, the hypotenuse was 2, a number the Greeks soon found was not a rational number. (It isn't the ratio of two integers!) The first reaction of the Pythagoreans was "don't tell anybody," a device still in use today by industry and governments. To save their philosophy, their next ploy was to declare that an irrational hypotenuse was made up of an infinite number of "tiny bits."
At this point the philosopher Zeno put the screws to the Pythagoreans with four paradoxes that easily arise when thinking about infinite collections. The age-old tale of the hare and the tortoise is actually one of these. By rational argument, the hare (Zeno actually used Achilles) could never pass the tortoise, but in reality it would. Hence, logically, an infinite process (like adding up the infinite number of terms in an infinite series) can't be carried out, but in reality, it possibly can.
Another of Zeno's paradoxes was that of the archer who shoots an arrow whose motion at each instant of time is "frozen." If the arrow is not moving at each of an infinite number of instants, then it cannot get anywhere. Zeno welcomed anyone who believed that argument to stand in front of the archer. So again, sloppy logic with an infinite process leads to one conclusion, but reality declares another outcome.
So, is that the only need for a theory of infinite collections? By the way, what is an infinite collection? Throughout the earlier chapters in this work, the symbol "∞" has been used to represent a quantity that is larger than any posited number. The whole of the calculus has been built with this understanding of "infinite." But in Euclidean geometry the student should have met the proposition that "the whole is equal to the sum of its parts" so "the whole is greater than any of its parts." If the even integers are "counted" by matching each of them (say, 2 k) with a counting number k, then there are as many even numbers are there are integers, and the whole is the same size as one of its parts. Infinite collections begin to become a lot more interesting!
Here is another confounding demonstration with the infinite. Let S=∑k=0∞−1k=1−1+1−1+⋯, and add this in the following two different ways.
Does this prove that S=0=1?
So, it becomes clear that there is a need for a theory of infinite collections and a set of rules by which such collections are manipulated. That is the real point of this chapter, to provide a framework for thinking about, and using infinite collections.
One type of infinite collection that has proven to be very useful in mathematics and its applications in the sciences is that of the infinite series, especially the infinite series of functions. In Sections 8.4 and 8.5, a glimpse of the utility of such series can be seen when just positive-integer powers of x are added. In the applications beyond this first calculus course, sums of sines and cosines, of polynomials, of the so-called "special" functions (Bessel, Heun, hypergeometric, etc.) turn out to be exceedingly important.
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