30 Nov 2008

Introducing the Infinite Series: Zeno's Paradox in Edwards & Penney

presentation of Edwards & Penney's work, by Corry Shores
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Edwards & Penney's Calculus is an incredibly-impressive, comprehensive, and understandable book. I highly recommend it.

In the fifth century B.C., Zeno proposes this paradox: in order for a runner to travel a given distance, she must first travel halfway, then half the remaining distance, than half that remainder, and so on ad infinitum (this description is a bit different but more illustrative than Zeno's paradoxes as recorded). But, because the runner cannot achieve infinitely many tasks in a finite period of time, motion from one point to another is impossible.

This paradox suggests the following subdivision of interval [0,1]:

Here there is a subinterval of length

for each integer n = 1, 2, 3, . . . . and so on. So if the total length of the interval would be the sum of all the subinterval's lengths into which the whole interval is divided, then:

Because, for example, 1/16 is the 4th term in the series, and 2 to the 4th power is 16; and, all these terms somehow add up to 1.

And yet, if we consider the formalization for the infinite series of integers:

It would seem that they do not add up to any finite value, even though the Zeno's paradox series does seem to add up to the finite value 1.

The study of the sums of infinite series taking the form

aims to determine in what sense such a sequence can have some sort of mathematical meaning.

from Edwards & Penney: Calculus. New Jersey: Prentice Hall, 2002, page 682a.

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