Elsewhere we proved that the sequence
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,
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
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 right-hand 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