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Given a positive real number $x,$ a sequence $\{a_n(x)\}_{n \geq 1}$ 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 $\lim\limits_{n \to \infty} a_n (x)$ exists.

How to tackle this problem? A small hint will be warmly appreciated.

Thanks a bunch.

$\textbf {Source} :$ NBHM PhD Screening Test $2021.$

RKC
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1 Answers1

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It seems to be a problem regarding an infinite tower of exponentials. Maybe checking this page and it's references you may approach something. https://mathworld.wolfram.com/PowerTower.html.

PS: I can not comment because haven't reached 50 rep. Thats why I post this as an answer and not just a comment.

FelipeCruzV10
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  • I have found in the link that the infinite tower converges if and only if $e^{-e} \leq x \leq e^{\frac {1} {e}}.$ But it isn't proved there. Do you know any proof of this fact? If so, could you please share some idea? Thanks in advance. – RKC Sep 11 '21 at 06:38
  • Not really, sorry. If i find something I'll let you know. – FelipeCruzV10 Sep 11 '21 at 06:41
  • I think I got it. At least the upper bound. – RKC Sep 11 '21 at 06:43
  • If the limit exists and equals to $y$ say then we have a functional equation $x^y = y$ or in other words $x = y^{\frac {1} {y}}.$ We need to maximize $x$ and hence we just need to compute the global maxima of the function $y \mapsto y^{\frac {1} {y}}.$ Isn't it so? Now if you compute the maxima it will be attained exactly at the point $y = e.$ Hence the largest possible value of $x$ for which the given limit exists turns out to be $e^{\frac {1} {e}}.$ – RKC Sep 11 '21 at 06:46
  • Sounds good. And it matches the result mentioned before. In this case it should be enough for your answer because you're looking for the largest value of $x$. – FelipeCruzV10 Sep 11 '21 at 07:14
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    Yes @FelipeCruzV10. BTW Thanks for your comments. – RKC Sep 11 '21 at 07:26
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    @RabinKumarChakraborty. I have a simple answer but for some completely unapparent reason your question has been closed. (?) – Oliver Kayende Sep 17 '21 at 16:49
  • @FelipeCruzV10. Why has this question been closed ? – Oliver Kayende Sep 17 '21 at 16:49
  • I don't know... – FelipeCruzV10 Sep 17 '21 at 17:25
  • I don't know. Some people in this website thought that they are the smartest persons in the globe. But actually they are bloody idiots. – RKC Sep 17 '21 at 19:33