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Possible Duplicate:
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.

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    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.2012-03-26

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