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Consider for positive real $x$ :

$$ f(x) = \sqrt x^{{\sqrt[3]{x}}^{\sqrt[4]{x},\cdots}} $$

How does this function behave ?

How fast does it grow ? Faster than any fixed iteration of exp sure, but How fast exactly ??

A brute estimate would be $\operatorname{tet}( \ln(x+1))$ but that would likely be a bad estimate. We have to do better.

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  • $\begingroup$ Could you use parentheses to make it clear which is a power of what? $\endgroup$ – Arnaud Mortier Feb 6 '18 at 18:17
  • $\begingroup$ It is a square root ^ Cuberoot etc ... I failed the root symbols Tex... $\endgroup$ – mick Feb 6 '18 at 18:19
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    $\begingroup$ So it's $$f(x) = (x^{1/2})^{(x^{1/3})^{(x^{1/4})^\cdots}}$$? $\endgroup$ – Antonio Vargas Feb 6 '18 at 18:23
  • $\begingroup$ Yes indeed Antonio $\endgroup$ – mick Feb 6 '18 at 19:57

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