<s>[INST] Evaluate of number represented by the infinite series $\sqrt{\tfrac{1}{3} + \sqrt{\tfrac{1}{3} + \sqrt{\tfrac{1}{3} + \cdots }}}$. Evaluate of number represented by the infinite series 
$$\sqrt{\tfrac{1}{3} + \sqrt{\tfrac{1}{3} + \sqrt{\tfrac{1}{3} + \cdots }}}$$
 [/INST] $n > 0:$
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 $n > 0$).
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