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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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### Tetration convergence: prove $\lim_{x\rightarrow0} {}^{n}x = \begin{cases} 1, & n \text{ even} \\ 0, & n \text{ odd} \end{cases}$

I'm a computer student, learning math just for fun. Today I was graphing for fun that I found something strange! I noticed that that wired function ${x^{x^{\cdot^{\cdot^{x}}}}}$ in zero, seems to ...
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### Convergence of the sequence $x_{n+1} = a^{x_n}$ [duplicate]

Let $a > 0$. Show that the sequence defined by $$x_0 = 1, \qquad x_{n+1} = a^{x_n}$$ converges for $a \leq e^{1/e}$. Any help is appreciated, I don't even know where to start with this. Edit to ...
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### Finding the derivative of $x$ tetrated to the $x$

Differentiating the functions $x^x$, $x^{x^x}$ (or ${^2{x}}$ and ${^3{x}}$), etc., although somewhat tedious, is pretty straightforward. I've even seen in a couple of books (and even on a post on this ...
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### What Tetration Algorithm can I utilize

I am trying to write program, that builds fractal, like a mandelbrot, but somewhat special... I could explain my "thought flow": A mandelbrot fractal (I call it "operation order 1") is done by ...
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### Interpolation between iterations of exponentials (and logarithms)

I'm interested in finding a continuous and ideally smooth family of real-valued monotonic functions $H_t(x)$ that interpolate between iterations of exponentials, and also iterations of logarithms. So ...
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### Is there a “natural tetration function”?

For the natural exponential function, $$f(x)=e^x \to f(x)=f'(x).$$ Is there a natural tetration (tetral?) function? $$f(x)={{^x}b} \to f(x)=f'(x)$$ Is it base $e$?
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### How to compute the indefinite integrate of n-th tetration of x?

How can the following indefinite integral be computed ? $\int{x↑↑n} dx$ where $n$ = {$x$ $\in$ $N^+$ : ${x > 2}$} Here ${x↑↑n}$ refers to $n$th tetration of $x$. I tried searching over the ...
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### Are there integer solutions to the equation ${^n}a+{^n}b={^n}c$?

A couple days ago, someone posted a question about using integer solution to the equation $a^a+b^b=c^c$ to disprove Fermat's last theorem. The question has since been deleted but I was curious as to ...
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### Under what “natural” combinatorial conditions would tetration or higher hyperoperations appear?

This is quite a soft question and is not the same as this question. I want to know what sort of "natural" problems one might examine in combinatorics whose solutions naturally require higher order ...
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### Exponent is to exponentiation as _______ is to tetration

Would it be tetrand, tetrant, or something else?
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### Bounds of fractional tetration

I know about Kneser, but if we take a simple recursion $$^{1/d}b=c, a(0)=b, a(n)=b^{\frac{1}{^{d-1}(a(n-1))}}$$ so $$\lim\limits_{n\to\infty}a(n)=c$$ and we can quickly find $c$ for positive ...
### Is there an equation $f(k,m,n)=n$ where $n$ is the number of applications of the Ackermann function needed?
Using the definition of the Ackermann function on page 247 of this paper, (sidenote, great paper):  \alpha(k,m,n) = \begin{cases} m+n, & k=1 \\ m, & n=1 \\ \alpha(k-1,m,\alpha(k,m,n-1)),&...