Nested Radicals

Let,

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

or,

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

Now,

Lets find the value of

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

or,

$2 + y = y^{2}$

On solving the quadratic equatio we get,

$y = 2 , -1$ (but y cannot be negative)

So,

$y = 2$

Therefore,

$x = \frac{1}{\sqrt{2}}\times2$

$x = \sqrt{2}$

The trick to this problem is to notice the the nested radicals grow at the same rate as the powers of the denominator such the each denominator $=\sqrt{2}$ and thus you can pull it out. Then we have the usual nested radical problem which is solved via $x=\sqrt{2+x}$ which gives x=2, then multiplying by the pulled out denominator gives $\sqrt{2}$