well. Again, by realizing that the whole expression appears inside itself, we are left with the equation If we solve this equation, we find that x = 2. More generally, we find that is the positive real root of the equation x3 − x − n = 0 for all n > 0. For n = 1, this root is the plastic number ρ, approximately equal to 1.3247. The same procedure also works to get as the real root of the equation x3 + x − n = 0 for all n > 1. Herschfeld's convergence theorem An infinitely nested radical (where all are nonnegative) converges if and only if there is some such that for all , or in other words Proof of "if" We observe that . Moreover, the sequence is monotonically increasing. Therefore it converges, by the monotone convergence theorem. Proof of "only if" If the sequence converges, then it is bounded. However, , hence is also bounded. See also Sum of radicals References Further reading Decreasing the Nesting Depth of Expressions Involving Square Roots Simplifying Square Roots of Square Roots Algebra