At the SouthWest Fox speaker dinner we were talking about continued (or infinite) fractions.

 

I posed this question:

 

What does this expression evaluate to?

 

 

       1 +            1

                  _________

                   1 +     1

                          ________

                           1 +   1

                                 _______

                                  1 +    1

                                         ______

                                          1 +    1

                                             

 

 

Clue: It’s adding some positive value to 1, so it must be larger than one. The thing it’s adding is the reciprocal of some number larger than 1 so….