Mistrust Intuition of the Infinite

The applet below illustrates a situation in which a sequence has a limit whose value does not seem to confirm to our geometric intuition.

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.

What if applet does not run?

Start with a square - a big square in the sequel. The midpoints of its sides form another square - called a small square below. On the sides of the latter form a sequence of broken lines as shown in the applet. Intuitively, the sequence of the so formed curves approaches the small square as the number of the "break points" tends to infinity. Let S be the perimeter of the big square, s that of the small square, and sn the total length of the broken line with n break points on a side of the small square. Since, obviously - and correctly in some sense that could be formalized - the broken line approaches the small square as n grows, one may expect that the same is true of their lengths:

limn→ ∞sn = s.

However, this is not true. Indeed, s = S/2, while, for every n > 0, sn = S. Since sn is constant, the limit of the sequence sn is that constant value:

limn→ ∞sn = S ≠ S/2 = s.

The example may make one wonder. Indeed, practically the same approach worked for estimating the circumference of a circle. Why does not it work now? In both cases, there is a sequence of curves convergent to a limit curve. In the case of the circumference, the lengths of the curves did converge to the length (circumference) of the limit curve (a circle). In the present case, the sequence of the lengths does not converge to the length of the limit curve (the small square.)

It appears that the convergence of the curves does not guarantee the (expected) convergence of their lengths.

Related material

  • Cavalieri's Principle
  • Derivative of Sine and Cosine
  • Distance From a Point to a Straight Line
  • Estimating Circumference of a Circle
  • Maximum Volume of a Cut Off Box
  • Naturally Discontinuous Functions
  • Rolle's and The Mean Value Theorems
  • Function, Derivative and Integral
  • Area of a Circle by Rabbi Abraham bar Hiyya Hanasi
  • Schwarz Lantern
  • Two Circles and a Limit
  • Deceptive Appearances
  • Problem 4010 from Crux Mathematicorum

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