I think this one goes to section of nested radicals, I was trying to solve if for a couple of days now. Maybe you have some nice solution to this one.
$$\sqrt{1\sqrt{2\sqrt{3\sqrt{4\sqrt{...}}}}}$$
I think it should converge. I computed with 1000 elements and 100000000 elements and got same result of 1.66169.
A more general case: $$g(n)=\sqrt{n\sqrt{(n+1)\sqrt{(n+2)\sqrt{(n+3)\sqrt{...}}}}}$$
Has a very nice result of: $$g(n+1)=\frac{g^2(n)}{n}$$
So solving simple case $g(1)$ would also solve more general one.
Edit: My question was if there was a smart way of calculating $\sqrt{1\sqrt{2\sqrt{3\sqrt{4\sqrt{...}}}}}$.