# 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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### How to find examples of periodic points of the (complex) exponential-function $z \to \exp(z)$?

Background: By considering the question which asks whether a certain summation-method $\mathfrak M$ for the (extremely divergent!) sum $\mathfrak M: S(z)=z + e^z + e^{e^z}+e^{e^{e^z}} + ...$ might be ...
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### Tetration value for x=0 [duplicate]

I came across a video about the function $f(x) =x^{x^{x^{x^{...} }} }$ and I played around with GeoGebra trying to graph it. I've observed that if the number of x's is odd, then $f(0)$ is always ...
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### Infinite tetration for complex domain [duplicate]

$$f(z)=z^{z^{z^{.^{.}}}}$$ $$for\space z \space \in C\rightarrow C$$ Can we define the range of this function(convergence-divergence specifically) without taking help from fractals? I checked out ...
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### Computation of Digits in Tetration [duplicate]

According to Wikipedia, $^44$ has $8.1 \cdot 10^{153}$ digits. How can I calculate the number of digits for an arbitrarily large tetration, such as $^{11}11$? Thank you!
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### Behaviour and closed forms of iterated functions

I'm interested in the behaviour of applying the same function repeatedly or oscillating between applying two different functions repeatedly. Let me explain. If I wanted to know what happens when I ...
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### Why is $x^{1.36^x}$ such a good approximate to $\int_{0}^{x}t^t dt$?

So, once again I was experimenting on Desmos and found that $\int_{0}^{x}t^t dt$ can be approximated pretty well by the function $x^{1.36^x}$. It roughly becomes more accurate as $x$ approaches to ...
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### Why tetration and more are not common in nature?

I didn't see this question in here but it was asked in quora and it was interesting to me that no one had any satisfying answers. some people suggested that it's because exponentiation describes the ...
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### Verifying uniqueness of my tetration

Previous two posts: Numerical instability of an extended tetration Verifying tetration properties Update: The first link only verifies continuity on $\mathbb R$, and so continuity cannot be used for ...
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### Numerical instability of an extended tetration

For bases $a\in(1,e^{1/e})$, ${}^na=a^{({}^{n-1}a)}=a^{a^{a^{.^{.^{.^a}}}}}$ converges to a value denoted as ${}^\infty a$. By observing the convergence rate of this sequence, we can derive the limit: ...
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### How can I find the value of a non-repeating exponent tower?

There are ways to express the function $$f(x)=x^{x^{x^{x^{x^{x^\cdots}}}}}$$ with $f(x)=\dfrac{W(-\ln(x))}{-\ln(x)}$ for other function like this; $$g(x) = x^{-x^{x^{-x^{x^{-x^{x^\cdots}}}}}}$$ I ...
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### Find $\lim_{x\rightarrow 0}x^{x^{x^x}}$ [duplicate]

I already known how to prove that $\lim_{x\rightarrow 0}x^{x^x}=0$ and $\lim_{x\rightarrow 0}x^x=1$. I also tried to use L'Hôpital's rule for this question but it didn't work. How to find the limit? (...
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### Find the maximum of $x^{x^{x^{⋰}}}.$

Question: Find the maximum of $x^{x^{x^{⋰}}}.$ Let $y = x^{x^{x^{⋰}}}.$ Then \begin{align} y & = x^y \\ \Rightarrow \ln y & = y\ln x \\ \Rightarrow \frac{1}{y} \frac{dy}{dx} & = y\left(\...
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### Tetrating by non-integers? [duplicate]

Recently, I've become interested in hyperoperations. I wondered what the equation y=x tetrated by x (xx), which is the same as x pentated by 2 (x2), would look like on a graph. To do this, ...
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### Rightmost decimal digits of Graham's number

How to find rightmost $n$ decimal digits of Graham's number efficiently. The last 500 digits are on the wiki/Graham's_number, but I want to know more. PowerTowerMod seems to be able to do it but is ...
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### How to find the maximum arc length of ${^{\infty}x}+{^{\infty}y}={^{\infty}r}$ and the value of $r$ at which it occurs?

After seeing a discussion about graphs of the relationship $x^x + y^y = r^r$, it got me interested in attempting to see what the graphical appearance of ${^{\infty}x}+{^{\infty}y}={^{\infty}r}$ would ...
### How to express the infinite power tower $a^{{{{{{(a+1)}^{(a+2)}}^{(a+3)}}^{.}}^{.}}^{.}}$?
Just a relatively simple question; I'm just wondering what would be the proper notation to use to express an infinite power tower that has each repeated exponent increasing by a value of $1$, like ...