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Homework Help: Infinite Exponent towers

  1. Jan 28, 2005 #1
    Infinite Exponent "towers"

    Please help me solve this problem; I dont even know how to start...

    Solve for x: [tex] x^{x^{x^{x^{...}}}}=2 [/tex]
     
    Last edited: Jan 28, 2005
  2. jcsd
  3. Jan 28, 2005 #2
    if x=1, x^x^x^x.... =1
    if x >1, x^x^x^x.... = infinity
    so x is undefine
     
  4. Jan 28, 2005 #3

    dextercioby

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    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 [itex] \mathbb{C} [/itex]... :wink:
     
  5. Jan 28, 2005 #4

    dextercioby

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    [tex] x^{x^{x^{x^{...}}}}=2 [/tex]

    Daniel.
     
  6. Jan 28, 2005 #5

    saltydog

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    Teteration

    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
     
  7. Jan 28, 2005 #6
    So far, I got that

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

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

    Go for it, I'll think about that, too.
     
    Last edited: Jan 28, 2005
  8. Jan 28, 2005 #7

    dextercioby

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    Try to see whether it verifies the equation...

    Daniel.
     
    Last edited: Jan 28, 2005
  9. Jan 28, 2005 #8
    It doesn't, [tex]\sqrt{2}^{\sqrt{2}^{\sqrt{2}}}[/tex] does infact come very close to 2 (I think exactly, dont have my 89 with me...), however. But as soon as more terms are piled on, it spirals into infinity.
     
  10. Jan 28, 2005 #9
    This is obvious for the expression [tex]((x^x)^x)^x...[/tex]
    but how did you show convergence for
    [tex]x^{x^{x^{x^\cdots}}}[/tex]
    ?
    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.
     
    Last edited: Jan 28, 2005
  11. Jan 28, 2005 #10
    I'm sorry, the correct expression for me would have been 0<x<1 will converge to 1
     
  12. Jan 28, 2005 #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.
     
  13. Jan 28, 2005 #12

    dextercioby

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    SQRT [2] IS THE CORRECT ANSWER...


    Generally

    [tex] x^{x^{x^{x^{...}}}}=a [/tex]

    Has the solution

    [tex] x=a^{\frac{1}{a}} [/tex]

    Iteration & logarithmation to show it...

    Daniel.
     
  14. Jan 28, 2005 #13
    I see where you are coming from, but I cant 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...
     
  15. Jan 28, 2005 #14

    dextercioby

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    No,it doesn't,trust me...Do you approximate results (intermediary) ??If so,then that's why it may jump over 2...

    Daniel.
     
  16. Jan 28, 2005 #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. :)
     
    Last edited: Jan 28, 2005
  17. Jan 28, 2005 #16
    I idnt approximate. I got 2 for a stack of 3 sqrt twos, but as I did [tex]x^{x^{x^{x^{...}}}}[/tex], it went to infinity.

    [tex]\log_{x}2=x^{x^{x^{x^{...}}}}[/tex]

    Can you show me how your solution was attained?

    I see the link, I'll go check it out.
     
  18. Jan 28, 2005 #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
     
  19. Jan 28, 2005 #18

    Hurkyl

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    vincent: a^a^a^a means

    [tex]
    a^{a^{a^{a}}}
    [/tex]

    not

    [tex]
    (((a^a)^a)^a)^a
    [/tex]


    Or, written flatly, it's x^x^x^x := x^(x^(x^x)))
     
    Last edited: Jan 28, 2005
  20. Jan 28, 2005 #19
    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)).
     
  21. Jan 28, 2005 #20
    so the expression is:
    x^(x^(x^(x^(x.....)??
    i was keep doing
    x^x^x^x^x......

    idoit me
    for your [tex]
    a^{a^{a^{a}}}
    [/tex]
    it really depend on how ppls read.....
     
  22. Jan 28, 2005 #21

    dextercioby

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    Anyways,the problem asked for the solution of the equation.Not for divagation regarding "power tower"...

    Vincentchan,are YOU having a bad day??

    Daniel.
     
  23. Jan 28, 2005 #22

    StatusX

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    [tex]\sqrt{2}[/tex] is obviously incorrect. It diverges, just like all x>1. I'm guessing you arrived at that answer by something like this:

    [tex] x^{x^{x^{x...}}} = 2[/tex]
    [tex] x^{x^{x^{x...}}} = x^2[/tex]
    [tex]x^2 = 2[/tex]
    [tex]x = \sqrt{2}[/tex]

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

    [tex]s = 1 + 2 + 4 + 8 + ...[/tex]
    [tex]2s = 2 + 4 + 8 + ... = s - 1[/tex]
    [tex]s = -1 [/tex]
     
    Last edited: Jan 28, 2005
  24. Jan 28, 2005 #23

    dextercioby

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    YOU ARE TERRIBLY WRONG!!!!!!!

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

    [tex] x^{x^{x^{x^{...}}}}=a [/tex]

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

    Daniel.
     
  25. Jan 28, 2005 #24

    Hurkyl

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    It's fairly easy to prove the sequence [itex]\sqrt{2} \uparrow \uparrow n[/itex] converges as [itex]n \rightarrow \infty[/itex]: it's a bounded, monotone sequence.
     
  26. Jan 28, 2005 #25

    learningphysics

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    Careful how you plug this into your calculator...

    Are you calculating:
    [tex]x^{x^{x^{x^{...}}}}[/tex]

    Or are you calculating

    [tex](((x^x)^x)^x)^x...[/tex]

    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:

    [tex]a_0=\sqrt{2}^\sqrt{2}[/tex]
    [tex]a_1=\sqrt{2}^{(a_0)}[/tex]
    [tex]a_2=\sqrt{2}^{(a_1)}[/tex]

    etc...
     
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