Can we find this infinite root in term of elementary function? Let $f(x)=\left(x+f(x+1)\right)^\frac{1}{x}$. What is the value of $f(2)$ ?
More precisely,  how to find the value of $$\sqrt{2+\sqrt[3]{3+\sqrt[4]{4+\cdots}}}~?$$

Thank you.
I got the answer as $\sqrt{3+2\sqrt[3]6}/\sqrt[6]{6}$.I took the fact that among $1,\sqrt[2]2,\sqrt[3]3,\sqrt[4]4,…\sqrt[n]n$,…to infinity ∛3  is the largest one. And I have used the best approximation for this.
I got the answer as $\sqrt{3+2\sqrt[3]6}/\sqrt[6]{6}$.I took the fact that among $1,\sqrt[2]2,\sqrt[3]3,\sqrt[4]4,…\sqrt[n]n$,…to infinity ∛3  is the largest one. And I have used the best approximation for this.