Does the infinite power tower $x^{x^{x^{x^{x^{x^{x^{.{^{.^{.}}}}}}}}}}$ converge for $x < e^{-e}$?

Question

As per the answer at https://math.stackexchange.com/a/573040/23890, the infinite power tower $x^{x^{x^{x^{x^{x^{x^{.{^{.^{.}}}}}}}}}}$ converges if and only if $ x \in [e^{-e}, e^\frac{1}{e} ] $.

Is $ e^{-e} = 0.065988... $ really the lower bound for $ x $ for this power tower to converge?

I wrote a Python program to check how this power tower behaves for $ x < e^{-e} $, for example, $ x = 0.01 $. It seems for $ x = 0.01 $, the power tower converges to $ 0.941... $.

from math import *

def power_tower_iterations(x, iterations):
    result = 1
    for i in range(iterations):
        result = x ** result
    return result

def power_tower(x):
    for iterations in [10, 100, 1000, 10000, 100000, 1000000]:
        print(power_tower_iterations(x, iterations))

power_tower(0.01)

Output:

0.9415490240371601
0.9414883685756696
0.9414883685756696
0.9414883685756696
0.9414883685756696
0.9414883685756696

So does the power tower converge for $ x < e^{-e} $ or does it not? If it does not, what error have I made above that led me to the false conclusion that the power tower converged to $0.941...$ for $x = 0.01$?

I think you are answered your question, the titled tower is converge iff $ x \in [e^{-e}, e^\frac{1}{e} ] $. — zeraoulia rafik, May 28 '20 at 16:34
your code only checks even iterates; if you did the computations on a hand calculator, you would see quickly that the sequence isn't settling at 0.941, but oscillating between 0.9414 and 0.01309 — Rob, May 28 '20 at 16:34
Does this answer your question? Surprising identities / equations — zeraoulia rafik, May 28 '20 at 16:36
@RobBland Thanks! If you post that as an answer, I can accept your answer and resolve this question. — Lone Learner, May 28 '20 at 16:41
Does this answer your question? Does the infinite power tower converge for all $0<x<1$ — Anthony Lahmann, Sep 18 '20 at 02:33

score 1 · Accepted Answer · answered May 28 '20 at 16:42

Hint :Nice answer is given here by @Simply ,For $0<x<e^{-e}$, it diverges. Note that for such small values of $x$, we get $t>x^{x^t}$ for when $t>y$ and $t<x^{x^t}$ when $t<y$, where $y=x^y$. In other words, adding more powers of $x$ ends up pushing it farther and farther away from $y$ instead of closer. Particularly, it approaches $0$ and $1$, for even and odd powers of $x$. In the same time limit of odd iteration is not the same with limit of even iteration for $0<x<e^{-e}$ as you showed in your code as @Rob Bland claimed above in comment

Does the infinite power tower $x^{x^{x^{x^{x^{x^{x^{.{^{.^{.}}}}}}}}}}$ converge for $x < e^{-e}$?

1 Answers1