Without using a computer, find \[\lim_{n\to\infty} \underbrace{\ln(\ln(\cdots\ln x))}_{n\text{ times}}\] for \(x\in\mathbb{R}^+\setminus\{1\}\)

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Without using a computer, find \[\lim_{n\to\infty} \underbrace{\ln(\ln(\cdots\ln x))}_{n\text{ times}}\] for \(x\in\mathbb{R}^+\setminus\{1\}\)

Calculus1
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Oh and if anyone's wondering, \(\mathbb{R}^+\setminus\{1\}\) is the set of all positive real numbers excluding \(1\).
ok i'll start by first conjecture, is it infinity ?

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It's not, it actually converges to a finite number in the complex plane.
oh wait as n goes to infinity i thought as x goes to :P
\[u = e^u\]
To be fair, I've been using Mathematica to find the limit, but I was wondering if there's an analytical method to finding a closed form (if it exists).
wolfram gives answer in terms of product log function
I suppose that's as closed as it's going to get :)
\[\lim_{n\to\infty} \underbrace{\ln(\ln(\cdots\ln x))}_{n\text{ times}} = -W_k(-1)\]
I was thinking there might be a closed form, let me try and show you how far I can get: \[y=\ln(\ln(\cdots))\] log both sides, the infinite side doesn't change, so that's just y still. \[\ln(y)=\ln(\ln(\cdots))=y\]\[y=e^y\]Do some algebra on it:\[-ye^{-y}=-1\] Here's the fun part that might be adjustable! We rewrite that right part: \[-1 = i*i = i*e^{i \pi}\] So now we have ALMOST something nice and invertible but not quite. \[-ye^{-y}=ie^{i \pi}\] Maybe there's something to do to play around with this to get a nice closed form I have some ideas give me a minute.
If it helps, the approximate value of the limit is \(0.318132+1.33724 i\).
Whoops I also made a mistake and wrote \(e^{i \pi}\) when it should be \(e^{i \pi/2}\)
We can see from this though: (using the slightly different \(-1=(-i)(-i)\) \[-ye^{-y}=-ie^{-i \pi/2}\approx-i\frac{\pi}{2}e^{-i \pi/2}\]\[y \approx i\frac{\pi}{2} \approx 1.57079i\ \approx 0.318132+1.33724 i\] So it's fairly close but not quite there. It seems to be multivalued though, I'm still playing with this it's interesting.
Yeah I guess as much as I wanna play around with this, it probably doesn't have a closed form since this also doesn't have a closed form: http://en.wikipedia.org/wiki/Omega_constant

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