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I came across this problem that I would like to ask you about:

For which values $a>0$ does there $\exists$ a limit of the sequence $$a, a^{a},a^{a^{a}}, a^{a^{a^{a}}}...$$

Well this looks like a recursive sequence. If $a=exp(\lambda)$ then $z_{0}=0, z_{n+1}=\lambda exp(z_{n}).$

and I guess for some $\lambda$ $\exists$ limit. I'm just not sure how to show for which.

After some trying out my idea was to check for the border case $$a> e^{1/e} $$

and $$a \leq e^{1/e}$$

but I guess both diverge. Any idea or input is greatly appreciated! Thanks

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    Euler's question? When did Euler ask this question?2012-11-15
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    Look up for "Shell-Thron-region" to have it for complex numbers2012-11-15
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    concerning Euler: what about "formulis exponentialibus replicatis" (Eneström 489) see: http://www.math.dartmouth.edu/~euler/pages/E489.html ?2012-11-15
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    There has been a long discussion about this here : http://math.stackexchange.com/questions/87870/are-the-solutions-of-xxxx-cdot-cdot-cdot-2-correct/87873#878732012-11-15

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