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# Questions tagged [tetration]

Tetration is iterated exponentiation, just as exponentiation is iterated multiplication. It is the fourth hyperoperation and often produces power towers that evaluate to very large numbers.

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### Super root function [closed]

Super root is an invertion of tetration. Lets define $f(x) = \sqrt[x]{x}_s$. Definition makes sense when x is integer. Is there an extension of this function to real numbers? Similar how Gamma ...
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### Proof of strictly increasing nature of $y(x)=x^{x^{x^{\ldots}}}$ on $[1,e^{\frac{1}{e}})$?

The title is fairly self explanatory: I have been trying to rigorously prove that $y(x)=x^{x^{x^{\ldots}}}$ is a strictly increasing function over the interval $[1,e^{\frac{1}{e}})$ for a while now, ...
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### The Physical Meaning of Tetration with fractional power tower

I've read a passage in the forum about tetration and did some research on Wiki. I understand the basic definition for any real height $n>-2$, $$^na=a^{a^{a^{a}}}...\text{, for real height =n}$$ ...
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### Continuum between addition, multiplication and exponentiation?

I noticed this old post which attempts to find the shades of grey between a linear and log scale where results are between zero and one. However, I was looking for the more general case where we find ...
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### A problem in understanding infinite towers (tetration)

To solve equations involving power towers (infinite tetration) we usually do something like this: $$x^{x^{x^{x^{\dots}}}} =k$$ $$x^{(x^{x^{x^{\dots}}})} =k$$ $$x^k=k$$ $$x=\sqrt[k]k$$ But what if ...
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### Algorithm for tetration to work with floating point numbers

So far, I've figured out an algorithm for tetration that works. However, although the variable a can be floating or integer, unfortunately, the variable ...
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### Inverse and named fixed values, with ↑↑?

The inverse of $+$ is $-$, of $\times$ is $/$ and of $\text{^}$ is Log. Continuing upwards hyperoperationally, what is the inverse of $↑↑$? Whats more somtimes values that are fixed are given names, ...
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### Last Digits of a Tetration

I was studying tetrations, or "power towers", and I found a decently well-known fact. The last $k-1$ digits of $^k 3 = 3^{3^{\vdots^{3}}} (k \text{ threes)}$ remain constant, for all numbers $^a 3$ ...
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### What number tetrated by itself equals a googol?

http://en.wikipedia.org/wiki/Tetration Tetrating stuff makes it really big really fast. I'm trying to figure out what number would equal a googol. Any help? Or a googolplex ?
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### Growth of $n!!\dots !$

The asymptotic growth of the factorial function $n!$ is famously given by Stirling's formula as $$n! \sim \sqrt{2 \pi n}\left(\frac{n}{e}\right)^n$$ Is there a similar formula for the iterated ...
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### Convergence of an infinite power

There are complex numbers $z$ and $w$ for which $$\lim_{n\rightarrow\infty}z\uparrow\uparrow n=w$$ where $\uparrow\uparrow$ is the tetration symbol, e.g. $z=\sqrt{2}$ and $w=2$. Are there complex ...
### How to differentiate $\lim\limits_{n\to\infty}\underbrace{x^{x^{x^{…}}}}_{n\text{ times}}$? [duplicate]
Let $$f(x)=\lim\limits_{n\to\infty}\underbrace{x^{x^{x^{...}}}}_{n\text{ times}}$$ Is it possible to find $f'(x)$. If yes, please show all steps.