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.
2026-03-25 14:03:24.1774447404
Can we find this infinite root in term of elementary function?
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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.