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Tower of power


bonanova
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Take a variable x and raise it to the xth power: xx; call the result y.
Now raise x to the y power: xy; call that result y. i.e. replace the previous value of y with the value of xy.
Again, raise x to the y power: xy; and replace the previous value of y with the value of xy.

Repeat this an arbitrarily large number of times, and set the result equal to 2.
That creates the infinite exponential sequence

x to the x to the x to the x to the x to the x .... = 2.

What value of x, if any, satisfies the equation?

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What this is saying is that the series will converge if the slope at the equilibrium is in the range (-1,1).

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What I meant is it will converge if started infinitesimally close to the equilibrium and the slope at the equilibrium is in the range (-1,1). The remaining math work is to show that it can get arbitrarily close to p(x) for x in the range (1/(e^e),e^(1/e)]...no matter where it is started.

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Sorry for resurrecting an old post, but I found some relevant math.

First, here are the links to where I found the information.

Tetration

Lambert W function

The inverse of the function y = x^x is x = (ln y) / W(ln y), where ln is the natural log (aka base e) and W is the Lambert W function.

The value of the Lambert W function can be approximated using Newton's method as follows (where wi is the ith appoximation in the series whose limit is the sought value). (Taken directly from wikipedia, w=W(z) so z = we^w)

wi+1 = wi - ( wiewi - z ) / ( wiewi + ewi )

The actual limit of xxxxx... = W(-ln x) / (-ln x)

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