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$$\sqrt{4 + \sqrt{4 + \sqrt{4 + \sqrt{4 + ........ \infty}}}} = $$
$$\frac{\sqrt{17} - 1}{2}$$
$$\frac{1 + \sqrt{17}}{2}$$
$$\frac{1 + \sqrt{13}}{2}$$
$$\frac{\sqrt{13} - 1}{2}$$
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