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

I depared myself with this problem


And studyins it I have made the assumption that if A is equal to a finite value then the expression is equivalent to
. 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 ?




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



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


1 Answer


Euler showed that

converges when
\le x
\le e^{1/e}

See this for discussion and references:

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