How to compute the following formulas? $\sqrt{2+\sqrt{2+\sqrt{2+\dots}}}$ 
$\dots\sqrt{2+\sqrt{2+\sqrt{2}}}$
Why they are different?

Suppose that the first converges to some value $x$.  Because the whole expression is identical to the first inner radical, $\sqrt{2+\sqrt{2+\sqrt{2+\cdots}}}=x=\sqrt{2+x}$ and solving for $x$ gives $x=2$.  Of course, I haven't justified that it converges to some value.
The second can be thought of as starting with $\sqrt{2}$ and repeatedly applying the function $f(x)=\sqrt{2+x}$.  Trying this numerically suggests that the values converge to 2.  Solveing $f(x)=x$ shows that $2$ is a fixed point of that function.
Looking at the second expression is actually how I'd justify (though it is perhaps not a rigorous proof) that the first expression converges.
