The General Infinite Exponential and Its Limit

Discussion in 'Math Research' started by I.N. Galidakis, May 26, 2008.

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    The iterated exponential x^x^...^x has been studied extensively, and its limit,
    whenever it exists, is known to be given by Lambert's W function, as h(x)
    = -W(-log(x))/log(x).

    Relatively little is known however, about the general infinite exponential
    a(1)^a(2)^...^a(n). I believe the following theorem to be a new result:

    Theorem:

    Whenever the general infinite exponential a(1)^a(2)^... converges, E m\in N:
    lim_{n->oo}(1)^a(2)^...^a(n) = a(1)^a(2)^...^a(m)^h(L), where L=lim_{n->oo}a(n).

    For a proof, please see:

    http://ioannis.virtualcomposer2000.com/math/exponents8.html

    My epsilon/delta proofs are a little rusty, so if anyone finds anything "funny"
    with the proof, please feel free to comment so I can correct it.

    Many thanks,
     
    I.N. Galidakis, May 26, 2008
    #1
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