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When you have:

$$x=y^{y^{y^{y\dots}}}$$

You have:

$$x=e^{-W(-\ln(y))}$$

ONLY when the power tower converges.

But what about when it doesn't? Is there any way to justify $2^{2^{2^{\dots}}}=e^{-W(-\ln(2))}\approx 0.82467854614-1.56743212384$?

Can we evaluate it in some analytical way or something when it doesn't converge, sort of like divergent summations and principal values?

Also, if you are simply going to argue that $2^{2^{2^{\dots}}}=\infty$, then I argue that I am asking about a way to make the evaluation non-infinite, much like how Ramanujan's summation method allows us to evaluate $1+2+3+\dots=-\frac1{12}$.

While it doesn't make "sense" because we see that it diverges to infinity, I was wondering if we could apply a similar method so as to get a finite value when the infinite power tower diverges to infinity.

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    $\begingroup$ It is possible to justify $e^{-W(-\ln(2))}\approx-0.571624 + 1.086461i$ with the extension of the Lambert's W(x) function in the complex domaine $(x<-e^{-1})$. But $2^{2^{2^{\dots}}}=e^{-W(-\ln(2))}$ is false because $2^{2^{2^{\dots}}}=\infty$. The Relationship between $y^{y^{y^{\dots}}}$ and $e^{-W(-\ln(y))}$ is derived from the equation $x=y^x$ wich is false if $y>e^{1/e}\approx 1.444668$ - $\endgroup$ – JJacquelin Feb 1 '16 at 10:09
  • $\begingroup$ @JJacquelin Thank you for your input, but I already knew most of that. I'll try and edit my question and see if you can add in some ideas. $\endgroup$ – Simply Beautiful Art Feb 1 '16 at 22:58
  • $\begingroup$ How did you arrive at the number $-0.571... + 1.086... i $? I tried your input at Wolfram Alpha and even several variants without getting even near to that value. $\endgroup$ – Gottfried Helms Mar 8 '16 at 7:27
  • $\begingroup$ @GottfriedHelms I think I used google's calculator. $\endgroup$ – Simply Beautiful Art Mar 8 '16 at 18:15
  • $\begingroup$ If you don't show a reliable/reproducable source for your curious constant $-0.57...+1.086...i $ or correct it I assume you're no more/not at all interested in that question and I'll downvote and propose to close. $\endgroup$ – Gottfried Helms Mar 11 '16 at 7:44
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I don't think that there can ever be a justification of a finite value for the notation $2^{2^{2^\cdots}}$ . But in fact in most cases, when we talk about tetration we see the things different, namely in form of an exponential tower, which begins with some value $x_0$ and puts an exponential base to it. So that, for the case of tetration, we should always write: $$ \begin{array} {r} \large x_0\\ \large {\,_2x_0 }\\ \large{\,_{\,_2}} \large {\,_2x_0 }\\ {\,_{\,_{\,_\ldots}}} \large{\,_{\,_2}} \large {\,_2x_0 }\\ \vdots \end{array}$$ (but which is difficult to model in $ \LaTex $)

With this we can write our beloved infinite towers and setting $x_0$ to the fixpoint near $\small 0.82467854 + 1.56743212 î $ $\,^{[1]}$ has then even a sensical interpretation (and reflects the infinite , let's call it now for distinction from the powertower, "exponential-tower").

$^{[1]}$ which is what also WolframAlpha gives for $e^{-W(-\ln(2))}$

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