Continued Square Roots
There is something interesting about the sequence
the sequence seems to be approaching the value 2. This can be proved by setting
Then
so that
is obviously not equal to -1 since all the terms of the sequence are positive (this is called a superfluous root) so
The sequence above is one of a family of sequences
Each of these sequence has a limit. et
where 
Then
so that
is obviously not equal to
since this is less than zero, so
