- 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
- 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.
- Hence, we say that, in the limit as , the complete sum S = 1.
- 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.
- 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 .