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Infinite tetration, convergence radius

Recently in this thread, Pseudo Proofs that are intuitively reasonable, I learned that $$\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{...}}}} = 2$$

The next natural question to ask is, what is the largest number $x$ such that $$f(x)=x^{x^{x^{x^{...}}}}$$ converges?

A short exercise in matlab coding suggests that either $f(1.5)$ diverges, or whatever it converges to is too large for my computer to handle. Thus the answer should be somewhere between 1.41 and 1.5.


marked as duplicate by Aryabhata, Qiaochu Yuan Mar 26 '12 at 20:15

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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    $\begingroup$ It is not explicitly mentioned on the linked question, but the radius of convergence you are looking for is $e^{1/e} = \max_y y^{1/y} \approx 1.4446678$. $\endgroup$ – TMM Mar 26 '12 at 20:18
  • $\begingroup$ Thanks for the answer TMM. Sorry to ask a duplicate question - I searched for "power tower" before asking, but did not know of the term "tetration". $\endgroup$ – Nick Alger Mar 26 '12 at 20:22
  • $\begingroup$ your expression could be reduced to an infinite product of (1/2)ln(x). Now use the infinite product convergence theorems and try compute rhs. sorry i am int rush to get to work would try post more details later. $\endgroup$ – Comic Book Guy Mar 26 '12 at 20:46