## anonymous one year ago Without using a computer, find $\lim_{n\to\infty} \underbrace{\ln(\ln(\cdots\ln x))}_{n\text{ times}}$ for $$x\in\mathbb{R}^+\setminus\{1\}$$

1. theopenstudyowl

impossible

2. anonymous

Oh and if anyone's wondering, $$\mathbb{R}^+\setminus\{1\}$$ is the set of all positive real numbers excluding $$1$$.

3. ikram002p

ok i'll start by first conjecture, is it infinity ?

4. anonymous

It's not, it actually converges to a finite number in the complex plane.

5. ikram002p

oh wait as n goes to infinity i thought as x goes to :P

6. ganeshie8

$u = e^u$

7. anonymous

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).

8. ganeshie8

wolfram gives answer in terms of product log function

9. anonymous

I suppose that's as closed as it's going to get :)

10. ganeshie8

$\lim_{n\to\infty} \underbrace{\ln(\ln(\cdots\ln x))}_{n\text{ times}} = -W_k(-1)$

11. Kainui

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.

12. anonymous

If it helps, the approximate value of the limit is $$0.318132+1.33724 i$$.

13. Kainui

Whoops I also made a mistake and wrote $$e^{i \pi}$$ when it should be $$e^{i \pi/2}$$

14. Kainui

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.

15. Kainui

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