How to evaluate the divergence or convergence of x to the power of X infinite times

I depared myself with this problem

xxxxxx…..=a{{{{{{{x}^{x}}^{x}}^{x}}^{x}}^{x}}^{…..}}=a

And studyins it I have made the assumption that if A is equal to a finite value then the expression is equivalent to
xa=ax^a=a
. studying this new function in terms of a , the function is increasing up to a=e , and after that point is decreasing .

How do I prove that there for all numbers greater than e , there arenآ´t any x that solves this equation ? How to study properly the convergence or divergence of this function?

Asking again ,how do I conclude that the function is diverging for X superior to e ?

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For xx superior to ee, can you say that the function is increasing?
– ذ°رپر‚ذ¾ذ½ ذ²ر–ذ»ذ»ذ° ذ¾ذ»ذ¾ر„ ذ¼رچذ»ذ»ذ±رچر€ذ³
2 days ago

  

 

I wasn’t entirely clear in my question . after the first assumption I begin to study the fuction f(a)=a^(1/a) , this function is increasing up to a= e ,And it has a maximum on that point (a,x) (e ,e^(1/e)).So ,if i prove that
– ricostynha
yesterday

  

 

so if i prove that the function x rise to x …..infinite times ,is increasiing after that point I prove taht is divergent after that point?
– ricostynha
yesterday

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1 Answer
1

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Euler showed that
xxx…x^{x^{x^{…}}}

converges when
e−e≤x≤e1/ee^{-e}
\le x
\le e^{1/e}
.

See this for discussion and references:

https://en.wikipedia.org/wiki/Tetration

I found this with
a Google search for
“euler exponential tower”.