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. etwhere

Then

so that

is obviously not equal tosince this is less than zero, so