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Given a positive real number $x$, a sequence $\{a_n(x)\}$ is defined as follows: $$a_1(x)=x\ \ \text{and}\ \ a_n(x)=x^{a_{n-1}(x)}\ \ \ \text{recursively for all}\ n\geq 2.$$Determine the largest value of $x$ for which $\lim_{n\to\infty}a_n(x)$ exists.

I observe that at $x=1$, sequence converges, whereas at $x=2$ sequence diverges. So $1\leq x<2$.

But I am unable to find the largest value. Is it somehow related with derivative? Please help.

PAMG
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This sequence is basically an infinite potency tower. If you look at the first few iterations you get $$ a_1(x)=x, a_2(x)=x^x, a_3(x) = x^{x^x}, ..., a_n(x) = x\uparrow \uparrow n $$ also known as Knuth's up arrow notation.

This post How can I prove the convergence of a power-tower? is probably what you are looking for. The maximum value is $e^\frac{1}{e}$

LegNaiB
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  • If you find an identical question with answers then you can flag the question as a duplicate instead of posting an answer which is essentially a link to another answer. – Martin R Aug 31 '21 at 15:19
  • Ah I didn't know about the flagging thing. Thanks for that info! But as what should I flag this question? Probably "in need of moderator intervention", right? – LegNaiB Aug 31 '21 at 15:20
  • You can flag it as a duplicate. – Martin R Aug 31 '21 at 15:22
  • I don't see that option. I only see "spam", "rude or abusive" or "in need of moderator intervention" as flagging options – LegNaiB Aug 31 '21 at 15:23
  • Perhaps because it is closed already. Otherwise there should be a "Flag -> a duplicate” option where you can enter the duplicate target. – With more reputation (>= 3K) the procedure changes: Then you can “vote to close” a question. – Martin R Aug 31 '21 at 15:27
  • Ok thanks for the detailed explanation! – LegNaiB Aug 31 '21 at 15:27