Find the limit of the following expression if it converges:
Also find a recurrence representation of the expression.
the limit is called Vijayaraghavan constant and it's almost 1.7579. see here for more details. what is fascinating about this constant
is that no closed form (in terms of other known constants) is known for Vijayaraghaven constant!! anyway, there's a theorem that
gives a necessary and sufficient condition for these kind of nested radicals to be convergent. here it is:
Theorem (Vijayaraghavan-Herschfeld): let and let then the
sequence is convergent if and only if the sequence is bounded.
so by the above theorem, your nested radical is obviously convergent. here's a more interesting, yet elementary, proof:
let clearly the sequence is positive and increasing. so if we prove that it's bounded, then we're
done! we'll do even better, i.e. we'll prove that to prove this claim, we'll define a new sequence: let and
call this an easy induction shows that call this result now fix an and define
we'll prove that the proof is by reverse induction:
if then by so which is the induction base! (remember, we're using reverse induction!) now suppose
the claim is true for and we'll prove it's also true for : so we have that thus and hence
by so: which completes the induction. now put to get