# Largest $x$ such that the power tower (tetration) $x^{x^{x^{x^{…}}}}$ converges? [duplicate]

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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 YuanMar 26 '12 at 20:15

• It is not explicitly mentioned on the linked question, but the radius of convergence you are looking for is $e^{1/e} = \max_y y^{1/y} \approx 1.4446678$. – TMM Mar 26 '12 at 20:18