The general infinite surd

Answer 1

Assume that k is positive (else the root of k will be a complex number), and the root is the positive square root.

Suppose this thing converges and the value it converges to is x. Then,

x^2 - k = x
So, x^2 - x - k = 0.
Solving, this has two solutions and one is negative. It can't be the negative one because everything is always positive by our assumption, so if x exists it must equal the positive root, i.e. x = (1+√(4k+1))/2.

It remains to show that the finite surds :√k,√(k+√k),... actually _do_ converge. Denote the n'th term by x_n. Observe x_n is increasing, so if we can show x_n is bounded above, we deduce convergence and the limit must be the x above.

Why is x_n bounded? Observe

(x_n)^2 - (x_n) < k (write out the left hand side)
Therefore, x_n < x because at x we have equality, x^2 - x = k.

So we're done!

If you drop the assumption and allow negative square roots and complex numbers, it's all going to depend on which of the two square roots you choose so it's not going to be well defined.