Chapter 1.15, Problem 31P

A tall tower of circular cross section is reinforced by horizontal circular disks (like large coins), one meter apart and of negligible thickness. The radius Of the disk at height n is $1/(n1nn)(n\ge 2)$.

Assuming that the tower is of infinite height:

1. Will the total area of the disks be finite or not? Hint: Can you compare the series with a simpler one?
2. If the disks are strengthened by wires going around their circumferences like tires, will the total length of wire required be finite or not?
3. Explain why there is not a contradiction between your answers in (a) and (b). That is, how is it possible to start with a set of disks of finite area, remove a little strip around the circumference of each, and get an infinite total length of these strips? Hint: Think about units—you can’t compare area and length. Consider two: (1) Make the width of each strip equal to one percent of the radius of the disk from which you cut it. Now the total length is infinite but what about the total area? (2) Try to make the strips all the same width; what happens? Also see Chapter 5, Problem 3.31 (b).