TITLE: Why does $\sqrt{n\sqrt{n\sqrt{n \ldots}}} = n$?
QUESTION [10 upvotes]: Ok, so I've been playing around with radical graphs and such lately, and I discovered that if the 
nth x = √(1st x √ 2nd x ... √nth x);

Then
$$\text{the "infinith" } x = x$$
Example: 
$$\sqrt{4\sqrt{4\sqrt{4\sqrt{4\ldots}}}}=4$$
Try it yourself, type calc in Google search, hit  then a number, such as $4$, and repeat, ending with $4$, (or press the buttons instead).
I'm a math-head, not big enough though, I think this sequence is divergent or convergent or whatever, too lazy to search up the difference.
However, can this be explained to me? Like how the Pythagorean Theorem can be explained visually.

REPLY [2 votes]: Assume that you iterated infinitely many times (can take a while), and observed a convergence to $n$.
One more iteration yields $\sqrt{n.n}=n$.