In a similar vein to What is the maximum value of this nested radical?, I'd like to share a similar nested radical, but this time with changing fractional powers.
What is the minimum value of $$f_\infty=\frac{x}{\sqrt{x-\sqrt[3]{x-\sqrt[4]{x-\cdots}}}}$$ where the radicals go up by one each time?
- Here is a plot of $f_{19}$. We can see that as $x\to 1^+$, $\min f_{19}\to 1.7186$ which is strange as the denominator can only take the binary values $0$ or $1$ at $x=1$. The curve is monotonically increasing from $1$ onwards, which is expected as the numerator dominates.
Actually, a simulation in PARI/GP up to $f_{100}$ yields a minimum value of around $1.718565$, which is somewhat close to $e-1$, although I strongly doubt that it will ever reach that value.
Note that $f_k$ is defined in $(1,\infty)$ for all positive integers $k$, but the curve swings wildly for $(-\infty,1)$. It is, of course, not a good idea to differentiate $f_\infty$ directly, but unfortunately we can't take $x$ and $\sqrt{x-\sqrt[3]{x-\sqrt[4]{x-\cdots}}}$ separately as both are increasing.
Another interesting question: Why is the minimum value of $f_k$ for large $k$ not equal to the expected $0,1$ or $\pm\infty$? Is it possible to manipulate $f_\infty$ so that L'Hopital can be used to find the value of $1.718\cdots$?
Related are
but neither have been solved as of now.
