Suppose that \(0< a< b\). Which of the following continued fractions is bigger? Why? \[\cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{a}}} \quad \textrm{or} \quad \cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{b}}}\]
Suppose the fractions are continued in the same way, then which is bigger in the following pair and why? \[\cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{4+\cfrac{1}{a}}}} \quad \textrm{or} \quad \cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{4+\cfrac{1}{b}}}}\]
Now compare: \[\cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{4+\dotsb+\cfrac{1}{99+\cfrac{1}{100+\cfrac{1}{a}}}}}}\] and the same thing with \(b\) in place of \(a\).
