Tower of power

[bonanova]

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

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?

[EventHorizon]

...

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

...

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.

[EventHorizon]

I thought I would look at the following equation (again using z and p(z) as opposed to x's to leave x for graphing),

y=(z^(z^x)-x)/p(z)-x, (obviously undefined at p(z))

ugg....I left off a set of parenthesis... the equation should be y=(z^(z^x)-x)/(p(z)-x)

[EventHorizon]

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 w_i is the ith appoximation in the series whose limit is the sought value). (Taken directly from wikipedia, w=W(z) so z = we^w)

w_i+1 = w_i - ( w_ie^w_i - z ) / ( w_ie^w_i + e^w_i )

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