Suppose 0<a<b. Which of the following continued fractions is bigger and why?


1 2+ 1 3+ 1 a

or
1 2+ 1 3+ 1 b

Suppose the fractions are continued in the same way, then which is the bigger in the following pair and why?


1 2+ 1 3+ 1 4+ 1 a

or the same thing with b in place of a.

Now compare:
1 2+ 1 3+ 1 4+…+ 1 99+ 1 100+ 1 a

and the same thing with b in place of a.