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 resultnow fix anand define

we'll prove that the proof is byreverse 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