$\sqrt[2]{1+\sqrt[3]{1+\sqrt[4]{1+\cdots}}}\approx 1.5176001678...$

While on chat, an interesting limit popped out: $$\sqrt[2]{1+\sqrt[3]{1+\sqrt[4]{1+\cdots}}}\approx 1.5176001678777188...\lt\phi$$ robjohn determined its value for fifty places, and Inverse Symbolic Calculator yields nothing.

Is there a closed form for this nice limit?

Solution 1:

Let $$r_n := 1+\sqrt[2]{1+\sqrt[3]{1+\sqrt[4]{\ldots+\sqrt[n]{1}}}}$$ (where I've added the $1$ at the beginning because it looks nicer that way and I wanted to). Then we have that $r_1=1$ is a root of the polynomial $p_1(x) = x-1$ and that (by recursion, if you want) $r_n$ is a root of $p_n(x) = p_{n-1}(x)^n-1$.

I suggest to study this sequence of polynomials instead of your complicated sequence of nested roots.

Edit: Removed the following wrong claim (after a nice counterexample by Pink Elephants):

Unfortunately my Galois theory is a bit rusty, and I don't have time right now to review the subject, but I think it would be pretty straightforward to show that the polynomial induce a sequence of field extensions over $\mathbb{Q}$ which have strictly increasing degree, which should imply (I think) that the limit of the sequence of the $r_n$, if it exists, is transcendental.