The first in a series is the radical

where the expression, i.e., the addition of radicals embedding even more radicals, is understood to continue for ever. The meaning of this expression may be made quite precise as a limit of an infinite sequence of numbers:

The sequence is monotone increasing. Indeed in passing from x_k to x_{k+1} we replace a 2 with 2 + \sqrt{2} , making x_k \lt x_{k+1} .

Naturally enough, the Monotone Convergence Theorem assures the existence of the limit x = \lim_{n \to \infty}x_n . To use the theorem we need to show that the sequence x_n is bounded. This is done by induction. We claim that \forall n\,x_n \le 2 .

Indeed, this is true for x_1 = \sqrt{2} . Assume also that, for some k , x_k \le 2 . Then x_{k+1} = \sqrt{2 + x_k} so that

implying x_{k+1} \le 2 and we are finished with induction.

Now that we know that the sequence x_k has a limit x we are also able to find its value. Indeed

squaring which we get a quadratic equation x^2 - x - 2 = 0 , with two solutions x = 2 and x = -1 . The latter, being negative, is clearly unsuitable and we are left with the identity: x = 2.

The result is quite pretty

The next step is to consider the more general expression

for some a \gt 0 . We'll have to separate two cases: a \gt 2 and a \lt 2 .