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I really like infinitely nested radicals so when I was messing around with them in Desmos, I noticed that the domain of $\sqrt{x-\sqrt{x!-\sqrt{(x!)!-\sqrt{((x!)!)!-\dots}}}}$ seemed to converge to $1<x<2$. Interested, I tried using averages of the integrals of partial parts of the equation (starting with $\sqrt{x}$, $\sqrt{x-\sqrt{x!}}$, and so on) to estimate the convergence of the infinitely repeating equation (assuming it actually converges) and got 0.871327768063. I only went to ten repetitions because I was doing it manually so I doubt the accuracy of this value.

However, I would really to have an exact form (though I doubt one exists) and I have no idea how to solve the integral and have been unable to find any relevant MSE questions I can look at. My only idea is that I would have to change the factorials into gamma functions. Any help is appreciated and thanks in advance.

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  • $\begingroup$ Have you proved convergence? $\endgroup$
    – Kenta S
    Commented Jun 6, 2023 at 1:45
  • $\begingroup$ No, how would I go about doing that? I think it appears to converge but that is obviously not enough. $\endgroup$
    – Dylan Levine
    Commented Jun 6, 2023 at 1:47
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    $\begingroup$ Note that $\Gamma(x+1)\leq x$ on the interval $[1,2],$ so the function is bounded above by $\frac{1}{2}(-1+\sqrt{1+4x}).$ $\endgroup$
    – MandelBroccoli
    Commented Jun 6, 2023 at 2:09
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    $\begingroup$ I'm going to go out on a limb and say finding a closed form is a completely hopeless endeavor. Determining the convergence of the integral seems like a more interesting problem. $\endgroup$
    – Christian E. Ramirez
    Commented Jun 6, 2023 at 6:07
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    $\begingroup$ @C-RAM math.stackexchange.com/questions/4829228/… My now slightly less naive self agrees with you about this integral being a hopeless endeavor. $\endgroup$
    – Dylan Levine
    Commented Dec 17, 2023 at 15:41

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