# The General Infinite Exponential and Its Limit

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

1. ### I.N. GalidakisGuest

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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

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