Suppose I give you the sequence of integers , and so on, where denotes the greatest integer less than or equal to x—in other words, it means to round down. So the first number in the sequence would be , the next number would be , and so on. Given this sequence, what can you learn about (assuming that you didn’t already know anything about it)?
A more general question: given the sequence of integers for k = 1,2,3…, what can you learn about r?
The answer has many interesting connections to the theory of irrational numbers and continued fractions.