# Questions tagged [power-towers]

For questions pertaining to power towers: expressions like a^(b^(c^d))), which result from iterated exponentiation. The "hyperoperation" tag may be appropriate, too.

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### Solving $a_{n+1} = x^{a_n}$ for various $x$ [duplicate]

(a) Consider a sequence defined by $a_0=1$ and $a_{n+1}=(\sqrt2)^{a_n}$ . Prove that limit exists and find it. (b) Show that the limit doesn't exist finitely if we replace $\sqrt2$ by $1.5$. What are ...
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### Exponential Power Tower

My question is- $$(4x)^{{{{\sqrt x}^{\sqrt x}}^ \cdots}^\infty}=0.0625$$ How to solve it? Options- (A)$2^{1/24}$ (B)$2^{1/48}$ (C)$4^{1/48}$ (D)$2^{1/96}$ I am confused how to solve this infinite ...
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### Is 2(6)3 (2↑↑↑↑3) equal to 2^65536? And if yes, is 2(n)3 equal to 2 to the power of how many time 2 is repeated in the power tower?

I am writing a paper for the last digits in a chain power of 2. I was wondering if 2↑↑↑↑3 is 2^65536. Beacouse 2↑↑↑3 is 65536 or 2^16 and is written as 2^2^...2^2 16 times and 2↑↑3 is 16 or 2^4 and is ...
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### Proving a modified Ackermann function using induction

So I have 2 functions: $$a(0,y) = y+ 1\\ a(x, 0) = a(x-1,1)\\ a(x,y) = a(x-1,a(x,y-1))$$ And: $$c(0,n) = 0\\ c(1,n) = n^2 + n +1\\ c(m,0) = c(m-1,1)\\ c(m,n) = c(m-1,c(m,n-1))$$ And then I have:...
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### Ackermann-Péter Function: Proof by induction, power tower

So I have this Ackermann function: $$A(0,y) = y+ 1\\ A(x, 0) = A(x-1,1)\\ A(x,y) = A(x-1,A(x,y-1))$$ And I have another function: $$d(n) = 2^{2^{.^{.^{.^{.^{2}}}}}}$$ while the height of the ...
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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 ...
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### How to evaluate the infinite sequence of 0^(-1)^(-2)^(-3)^(-4)^{.}^{.}^{.}?

(Note: I deliberately didn't write the title in Latex because the notation doesn't display properly with power towers. Specifically regarding the ellipsis, which displays as "..." instead of ".^{.}^{.}...
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### Determine the smallest positive integer $m$ for which $\underbrace{100^{100^{\ldots^{100}}}}_m>\underbrace{3^{3^{\ldots^3}}}_{100}$

The functions $f$ and $g$ are defined by $f (x) = 3 ^ x$ and $g (x) = 100 ^ x$. Two sequences $a_1, a_2, a_3, \ldots$ and $b_1, b_2, b_3, \ldots$ are then defined as follows: (i) $a_1 = 3$...
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### 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 ...
287 views

### Area enclosed by a function containing two power towers: $f(x)=(−\ln(x↑↑(2k)))↑↑(2k+1)$

I've been considering functions involving power towers lately and come across the following function: $$f(x)=(−\ln(x↑↑(2k)))↑↑(2k+1)$$ $$\text{Where }k∈\mathbb{Z} ^+$$ In the image below we can see ...
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### What will be the minimum value of the function $f(x)=\arcsin(x)\uparrow\uparrow(2k)$ as $k \to \infty$?

Consider the following two functions involving power towers; $$f(x)=\arcsin(x)\uparrow\uparrow(2k)$$ $$g(x)=\arcsin(x)\uparrow\uparrow(2k-1)$$ Where $k\in \mathbb{Z}^+$. The global minimum of $f(x)$ ...
### 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 ...