Evaluate of number represented by the infinite series $\sqrt{\tfrac{1}{3} + \sqrt{\tfrac{1}{3} + \sqrt{\tfrac{1}{3} + \cdots }}}$.

Question

Evaluate of number represented by the infinite series

$$\sqrt{\tfrac{1}{3} + \sqrt{\tfrac{1}{3} + \sqrt{\tfrac{1}{3} + \cdots }}}$$

Answer 1

$n > 0:$

Let $x = \sqrt{n + \sqrt{n + \sqrt{n + \cdots }}}$

Assuming convergence:

$$\begin{align*} x = \sqrt{n + \sqrt{n + \sqrt{n + \cdots }}} & \implies x^2 = n + \sqrt{n + \sqrt{n + \sqrt{n + \cdots }}} = n + x \\ & \implies x^2 - x - n = 0\end{align*}$$

This is a simple quadratic to solve (take the positive root as $n > 0$).

Answer 2

HINT:

Let $a = \sqrt{1/3 + ...}$

Then $a = \sqrt{1/3 + a}$

Answer 3

Set x equal to this expression and square both sides. Subtract 1/3, and you have x again. Then solve this quadratic equation. Decide which of the roots is the correct answer