# Bisection

- Author:
- Ryan Hirst

- Topic:
- Limits

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 .

*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 (n*th*) term.