Infinite Sums of Iterated Radicals

The Problem:
	Compute and justify your answer.

The solution:

Answer. Let . Because x is defined by means of infinitely many square roots, it is not obvious that it represents a real number; however, should it be real then it would certainly be positive (because the square root symbol stands for the positive square root), and it would satisfy

That is, 0 = x² - x - 56 = (x - 8)*(x + 7), so that x would necessarily equal 8. In the same way, letting y =

should y represent a real number then it would be the positive number that satisfies

so that y² + y - 56 = (y - 7)(y + 8), whence y = 7. Combining these two calculations, we deduce that if both x and y are real numbers, then

Further comments. Many of our correspondents remarked that 56 could be replaced by any real number r with r > 1:

Moreover, in a recent article Javier Peralta ["On Some Infinite Sums of Iterated Radicals", International Journal of Mathematical Education in Science and Technology, 40:2 (March 2009) 290-300] claims to prove (but his proof seems to be missing a key step) that for r > 1,

is an integer k if and only if r = k(k – 1),

while

is an integer k if and only if r = k(k + 1).

Any of the methods discussed here would lead to a proof of Peralta's assertion.

math central