Continued Square Roots

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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