Nested radicals
Elsewhere we proved that the sequence
(1) 
is convergent to 2. With this in mind, it is possible and meaningful to shorten the description of this fact to
Expressions like that serve an example of infinite nested radicals. They are realized in terms of limits, if those exist, and are understood as the shorthand for the latter.
For a real s > 0,
(2) 
always exists. For s < 2, this follows immediately from the Monotone Convergence theorem because then sequence (2) is bounded by the sequence (1) termwise. For
The curious observation about (2) is that, like in (1), the infinite nested radical may equal an integer. In general, let
Then, adding s to both sides,
Thus we obtain a relationship between s and S: and using the quadratic formula also The negative number ought to be recognized as spurious and discarded. So, finally we have
on the one hand and
(3) 
on the other. From (3) we get several examples:

and so on.
An interesting quandary arises when we pick which would require and thus appear to imply
which, obviously could not be right. Indeed, the righthand side represents the limit of the sequence
with all terms zero. The sequence converges to 0, not 1! How does one explain that?
How do you explain
At one point we derived the relation and dismissed one solution with the sign minus as being negative and therefore unsuitable. However, for s = 0, the two solutions are i.e., 1 and 0. In this case, 1 is obviously spurious and should be dropped, leaving the correct identity
References
 C. A. Pickover, A Passion for Mathematics, John Wiley & Sons, 2005, p. 96
 C.C. Clawson, Mathematical Mysteries, Plenum Press, 1996, pp. 140144
Nice trick there with the spurious 1. I feel like this well deserves a place with all the other famous "1=0 contradictions" where the student is required to find the flaw.
August 2nd, 2010 at 3:12 pmYes, I agree.
In addition, this is a sort of interplay between algebra and analysis that is often disregarded, e.g., in questioning whether .999... = 1 or not. Several math notations include ellipses and have no meaning, except as limits. Once this is understood, algebra may be be fruitfully employed.
August 2nd, 2010 at 4:04 pmThe more obvious way to get S^2 = s + S is to just square the equation.
August 20th, 2010 at 4:23 amHubert, hi.
You are probably right, but should everything be always done the most obvious way? May there be a virtue in a little obfuscation so as to trigger a thought process. I am asking that because all my life  which probably means a few dozen times  I've been squaring the identity as the first step. This is the first time that I did it differently and here you at the right moment in the right place.
August 20th, 2010 at 4:37 am