Nested Radicals with multiplication

Question

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 $\sqrt{1\sqrt{2\sqrt{3\sqrt{4\sqrt{...}}}}}$.

Answer 1

Since $g(1) = 1^{1/2} 2^{1/4} 3^{1/8} \cdots$, we can write $\log g(1)$ as an infinite sum. $$ \log g(1) = \sum_{k=1}^{\infty} \frac{\log k}{2^k}. $$

This series obviously converges since $\log k < 1.5^k$, but I doubt it would be expressed in terms of elementary functions. (See this)

Answer 2

This is Somos' quadratic recurrence constant $($see also$)$, for which no closed form is yet known. Its decimal expansion can be found here. A similar expression, also lacking a known closed form, is the nested radical constant, whose digits can also be found on OEIS. Hope this helps.