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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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### 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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### Simplifying a 'fractal-like' expression with tetration

Let $f_2(n)=2^n n$ and let $f_3$ be defined recursively as $$f_3(n)=\underbrace{f_2\cdots f_2}_{n\text{ times}}(n)=f_2^n(n).$$ This will lead to tetration, but is it possible to write $f_3$ in a ...
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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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### Tetration and Fractions

Recently I discovered Tetration, and was wondering about having tetration with fractional "tetronents", take the example $$^{7/2}3\;\Bbb{or}\;3\uparrow\uparrow{\frac72}$$Initially it seems difficult ...