Nested Radicals

Firstly,

Let,

$x = \sqrt{ \frac{2} {2^{2}} + \sqrt{ \frac{2} {2^{4}} + .......} }$

Therefore,

$x = \frac{1} {2} \sqrt{ 2 + \sqrt{ 2 + .......} }$

Now lets find the value of,

$\sqrt{ 2 + \sqrt{ 2 + .......} } = y$

$y^{2} = 2 + \sqrt{ 2 + \sqrt{ 2 + .......} } = 2 + y$

By solving it we get $y = 2$

Therefore,

$x = \frac{1} {2} \times 2 = 1$

Taking for granted that the expression is meaningful, let X denote its value. Multiplying by 2, and pulling the factor inside the square roots, 2X= Sqrt[2 + Sqrt[2 + Sqrt[2 + ...]]]. Thus, 2X = Sqrt[2 +2X], so by the basic technique for nested radicals X=1.