Ryan Hirst
At each step, there is a sum (green) and a remainder (red). For the next step, the amount added to the sum and the amount left over are equal. In algebra, if we have the sum Then
  1. As k increases, the sum approaches 1: Given any finite number x, no matter how small, we can name the step when the red bar comes closer to B than x. In other words, there exists no finite amount which will remain red, as long as we agree never to stop subdividing.
  2. Hence, we say that, in the limit as , the complete sum S = 1.
  3. No tricks: By saying the sum (in the limit) S = 1, we respect the given information. The given length AB=1, and we divide in such a way that, provided we agree never to stop dividing, no amount however small can be left over.
  4. if we were confronted with a finite number of terms: , an infinite number of terms are discarded. Nevertheless, the sum of all these missing terms is .
If we are given a sum whose value we do not know, we can use bisection as a frame of reference. For example, if each term in the sequence is less than half the term before it ... then if we agree to add n terms, and then stop, the discarded amount (the error) is smaller than the last (nth) term.