What is the solution for x in the infinite exponent tower x^x^x^...=2?

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The solution to the infinite exponent tower equation \( x^{x^{x^{\cdots}}} = 2 \) is \( x = \sqrt{2} \). For \( x = 1 \), the expression converges to 1, while for \( x > 1 \), it diverges to infinity. The discussion highlights that the equation does not admit a real solution for values of \( x \) greater than \( e^{1/e} \). The convergence of the sequence generated by \( \sqrt{2} \) is established, demonstrating that it approaches 2 as more terms are added.

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Infinite Exponent "towers"

Please help me solve this problem; I don't even know how to start...

Solve for x: x^{x^{x^{x^{...}}}}=2
 
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if x=1, x^x^x^x... =1
if x >1, x^x^x^x... = infinity
so x is undefine
 
vincentchan said:
if x=1, x^x^x^x... =1
if x >1, x^x^x^x... = infinity
so x is undefine

You can say that the equation does not admit a real solution...

Daniel.

P.S.The problem would be interesting to consider and solve in \mathbb{C}... :wink:
 
Teteration

The Divine Zephyr said:
Please help me solve this problem; I don't even know how to start...

Solve for x: ((((((x^x)^x)^x)^x)^x)^ ...)=2

LaTeX can't seem to stack exponents, like they should be
(x^x^x^x^x^x^x^x^x^x^x^x^...)

Let the iterated exponent (a teteration) be called LHS (left hand side)

Then LHS=y=2
but the iterated exponent is also equal to LHS so that:
LHS^y=2

but y=2
so that:

x^2=2
or:

x=Sqrt[2]

Yea, I know it's hard to grasp. I need to work on it too.

SD
 
vincentchan said:
if x=1, x^x^x^x... =1
if x >1, x^x^x^x... = infinity
so x is undefine

So far, I got that

if x=1, my LHS=1
if 0<x<1, LHS converges to 1
if x>1, LHS diverges

Let the iterated exponent (a teteration) be called LHS (left hand side)

Then LHS=y=2
but the iterated exponent is also equal to LHS so that:
LHS^y=2

but y=2
so that:

x^2=2
or:

x=Sqrt[2]

I plugged it in on a calculator and it divirged into infinity...

P.S.The problem would be interesting to consider and solve in \mathbb{C}... :wink:
Go for it, I'll think about that, too.
 
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saltydog said:
x=Sqrt[2]

Yea, I know it's hard to grasp. I need to work on it too.

Try to see whether it verifies the equation...

Daniel.
 
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It doesn't, \sqrt{2}^{\sqrt{2}^{\sqrt{2}}} does infact come very close to 2 (I think exactly, don't have my 89 with me...), however. But as soon as more terms are piled on, it spirals into infinity.
 
The Divine Zephyr said:
if x<1, LHS converges to 1

This is obvious for the expression ((x^x)^x)^x...
but how did you show convergence for
x^{x^{x^{x^\cdots}}}
?
For the first, you get the sequence 2^{-1}, 2^{-1/2}, 2^{-1/4},...,2^{-1/2^i} but for the second the form gets ugly:
2^{-1}, 2^{-1/2}, {sqrt{2}/2}^sqrt{2}, {1/2}^{{sqrt{2}/2}^sqrt{2}}, ...
My forays with Windows calculator are giving me oscillations, so I can't be sure it doesn't converge to something less than 1.
 
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  • #10
hypermorphism said:
This is obvious for the expression ((x^x)^x)^x...
but how did you show convergence for
x^{x^{x^{x^\cdots}}}
?
For the first, you get the sequence 2^{-1}, 2^{-1/2}, 2^{-1/4},...,2^{-1/2^i} but for the second the form gets ugly:
2^{-1}, 2^{-1/2}, {sqrt{2}/2}^sqrt{2}, {1/2}^{{sqrt{2}/2}^sqrt{2}}, ...
My forays with Windows calculator are giving me oscillations, so I can't be sure it doesn't converge to something less than 1.

I'm sorry, the correct expression for me would have been 0<x<1 will converge to 1
 
  • #11
2^{-1} is less than 1. :) But I'm not sure the power tower sequence it generates converges to 1, as I can't find a general ith term for the sequence.
 
  • #12
SQRT [2] IS THE CORRECT ANSWER...


Generally

x^{x^{x^{x^{...}}}}=a

Has the solution

x=a^{\frac{1}{a}}

Iteration & logarithmation to show it...

Daniel.
 
  • #13
I see where you are coming from, but I can't see it working the the equation...


SQRT [2]^SQRT [2]^SQRT [2]=2, but as soon as more SQRT [2]s are stacked, it flies off the mark...
 
  • #14
No,it doesn't,trust me...Do you approximate results (intermediary) ??If so,then that's why it may jump over 2...

Daniel.
 
  • #15
Hmm. Mathworld has an expression for the general solution of the infinite power tower at http://mathworld.wolfram.com/PowerTower.html , but its not as simple as a^{1/a}. However, the equation seems to verify that the power tower of sqrt(2) converges to 2.
Regarding dex, it's true. sqrt(2) is an irrational number so it can't be rationed about like a finite decimal on a calculator. :)
 
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  • #16
I idnt approximate. I got 2 for a stack of 3 sqrt twos, but as I did x^{x^{x^{x^{...}}}}, it went to infinity.

\log_{x}2=x^{x^{x^{x^{...}}}}

Can you show me how your solution was attained?

I see the link, I'll go check it out.
 
  • #17
dextercioby:
are you having a bad day?? ...I'll show you if x=2^1/2, then x^x^x^x >2,

let x=2^1/2

x^x^x^x=x^(x*x*x) = x^(2*x) = 2^(1/2*2)^x = 2^x >2
 
  • #18
vincent: a^a^a^a means

<br /> a^{a^{a^{a}}}<br />

not

<br /> (((a^a)^a)^a)^a<br />


Or, written flatly, it's x^x^x^x := x^(x^(x^x)))
 
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  • #19
vincentchan said:
dextercioby:
are you having a bad day?? ...I'll show you if x=2^1/2, then x^x^x^x >2,

let x=2^1/2

x^x^x^x=x^(x*x*x)
This step is wrong. We're finding the result of x^(x^(x^...)), not ((x^x)^x)^...
Ie., (sqrt(2)^sqrt(2))^sqrt(2) = sqrt(2)^(sqrt(2)*sqrt(2)) = sqrt(2)^2 = 2,
but there is no similar way to simplify sqrt(2)^(sqrt(2)^sqrt(2)).
 
  • #20
so the expression is:
x^(x^(x^(x^(x...)??
i was keep doing
x^x^x^x^x...

idoit me
for your <br /> a^{a^{a^{a}}}<br />
it really depend on how ppls read...
 
  • #21
Anyways,the problem asked for the solution of the equation.Not for divagation regarding "power tower"...

Vincentchan,are YOU having a bad day??

Daniel.
 
  • #22
\sqrt{2} is obviously incorrect. It diverges, just like all x>1. I'm guessing you arrived at that answer by something like this:

x^{x^{x^{x...}}} = 2
x^{x^{x^{x...}}} = x^2
x^2 = 2
x = \sqrt{2}

The problem is that you are assuming \infty=\infty (since the power tower diverges), and this isn't always true. By the same logic, I could say:

s = 1 + 2 + 4 + 8 + ...
2s = 2 + 4 + 8 + ... = s - 1
s = -1
 
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  • #23
StatusX said:
\sqrt{2} is obviously incorrect. It diverges, just like all x>1. I'm guessing you arrived at that answer by something like this:

x^{x^{x^{x...}}} = 2
x^{x^{x^{x...}}} = x^2
x^2 = 2
x = \sqrt{2}

The problem is that you are assuming \infty=\infty, since the power tower diverges, and this isn't always true.By the same logic, I could say:

s = 1 + 2 + 4 + 8 + ...
2s = 2 + 4 + 8 + ... = s - 1
s = -1


YOU ARE TERRIBLY WRONG!

Please,do not GUESS WHAT I AM THINKING... :devil:

x^{x^{x^{x^{...}}}}=a

THIS EQUATION HAS THE SOLUTION i specified in the post with lots of red...

Daniel.
 
  • #24
It's fairly easy to prove the sequence \sqrt{2} \uparrow \uparrow n converges as n \rightarrow \infty: it's a bounded, monotone sequence.
 
  • #25
The Divine Zephyr said:
I idnt approximate. I got 2 for a stack of 3 sqrt twos, but as I did x^{x^{x^{x^{...}}}}, it went to infinity.

\log_{x}2=x^{x^{x^{x^{...}}}}

Can you show me how your solution was attained?

I see the link, I'll go check it out.

Careful how you plug this into your calculator...

Are you calculating:
x^{x^{x^{x^{...}}}}

Or are you calculating

(((x^x)^x)^x)^x...

The first one goes to 2 if x=sqrt(2). The second one does not.

The way to plug it into the calculator is like this:

a_0=\sqrt{2}^\sqrt{2}
a_1=\sqrt{2}^{(a_0)}
a_2=\sqrt{2}^{(a_1)}

etc...
 
  • #26
youre right, I'm wrong (those big letters are annoying, though). It seems like it would diverge for x>1, but it's only for x>e^(1/e). Which lead to the interesting paradox that x goes to 1 as a goes to infinity, and yet for x=1, a=1.
 
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  • #27
StatusX said:
(those big letters are annoying, though).

They were meant to be annoying,not for the fact that u contradicted me when you were wrong,and i was right,but for doing that by assuming what i was thinking... :rolleyes:

Daniel.
 
  • #28
Proof for convergence

Would anyone be willing to provide a reasonably short (if possible) proof for the convergence interval for the function? Or perhaps someone can cite an on-line reference for the proof. If I obtain one I can understand, I'll post it.

Thanks,
SD
 
  • #29
The solution to the equation
x \upuparrows \infty = a

is a^{\frac{1}{a}}
,which is never bigger than e^{\frac{1}{e}}

Daniel.
 
  • #30
dextercioby said:
The solution to the equation
x \upuparrows \infty = a

is a^{\frac{1}{a}}
,which is never bigger than e^{\frac{1}{e}}

Daniel.

Thanks Daniel,
However, my understanding is that the hyperexponent function above converges only when x is in some interval. Can you please tell me how to incorporate math symbols into my posting so I can be more specific? I have Mathematica. Can I export data from there?

Thanks,
SD
 

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