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Tetration Questions

  1. Dec 29, 2012 #1
    Hello all, of late I have been very interested in tetration (http://en.wikipedia.org/wiki/Tetration). I have a few questions that weren't really explained clearly by this wikipedia article. P.S., I can't get latex to work with the wikipedia notation, so I will use this notation: ##x^{\Delta n}##.

    1. Why does
    [tex]\lim_{n\to\infty} x^{\Delta n}[/tex] only converge for [tex]1/e<x<e^{1/e}[/tex]

    2. How can the exact value be calculated as a limit? For example, how could I prove that:
    [tex]\lim_{x\to\infty} \sqrt{2}^{\Delta x}=2[/tex]

    3. Why is the following statement true (from the wikipedia article):
    [tex]\lim_{x\to0}\lim_{n\to\infty}x^{\Delta n}=1/e[/tex]

    I will be sure to post more questions as they come.
     
    Last edited: Dec 29, 2012
  2. jcsd
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