There's a Limit

Explore the continued fraction: 2+3/(2+3/(2+3/2+...)) What do you notice when successive terms are taken? What happens to the terms if the fraction goes on indefinitely?

Not Continued Fractions

Which rational numbers cannot be written in the form x + 1/(y + 1/z) where x, y and z are integers?

Comparing Continued Fractions

Which of these continued fractions is bigger and why?

And So on - and on -and On

Stage: 5 Challenge Level:
Let $$f_{0}(x)= \frac{1}{1-x}$$ and $$f_{n}(x)=f_{0}(f_{n-1}(x))$$ where $n = 1,2, 3, 4, ...$

Evaluate $f_{2000}(2000)$