# 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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### About the inequality $f(-x)<\left(\Gamma(1-x)\right)^{\frac{1}{1-x}}$

Prove or disprove that $-1< x<0$ then we have : $$f(-x)<\left(\Gamma(1-x)\right)^{\frac{1}{1-x}}$$ Where : $$f(x)=2-x^{-\frac{212}{1000}x^{x^{\frac{1}{5}x^{2x^{\frac{1}{5}x}}}}}$$ My ...
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### Can $\int\limits_0^\infty e^{ix^ x}dx$ be written without a limit?

We know about the Fresnel Integrals: $$C(x)=\int \cos x^2 \, dx,\quad S(x)=\int \sin x^2 \, dx$$ which can also be written as: $$\int e^{ix^2}dx=C(x)+i\,S(x)$$ To make a more interesting and tetration ...
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### Integral form(s) of a general tetration/power tower integral solution: $\sum\limits_{n=0}^\infty \frac{(pn+q)^{rn+s}Γ(An+B,Cn+D)}{Γ(an+b,cn+d)}$

In many tetration/power tower integrals, one sees a general form of the following. Let this new function be notation used to show the connection between the general result and special cases using ...
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### Is there any formula to sum the series $\sum_{k=1}^n{2^{k^k}}$

$$\sum_{k=1}^n{2^{k^k}}$$ I couldn't find anything on this with a simple Google search so is it not trivial? If yes then can someone link some resources/Wikipedia page where I could read more? (can't ...
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### Integrating an Infinite exponent tower

$$\int_0^1 x^{2^{x^{2^{x^{\ldots}}}}} ~~ dx = ~~?$$                                                                                                                                           What I'...
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### Is it correct for $\int {x^x}\, dx$?

Recently, I was working with power towers. I was interested in $x^x$; it is easy to know its derivative, but I wanted to find its integral. Here's what I found. Please let me know if it is right or ...
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### Area under $\frac{1}{x^x}$ curve

How can I calculate area under $\frac{1}{x^x}$ on any interval, I tried the Archimedes method, but I get $$\frac1n\sum \frac 1{X_n^{X_n} }$$ and that's very complex to calculate because of the roots, ...
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### Solution verification: Derivative of the infinite power tower $y(x) = x^{x^⋰}$

I was doing the problem $(x^{x^{⋰}})'$ and I would like someone to verify my solution: \begin{align*} &\left(y=x^{x^{^{⋰}}}\right)'\\ \implies & \;\left(y=x^{y}\right)'\\ \implies & \;\...
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### What is 333^333^333 in mod 17

Is there an easier way to do this than finding cycles of different mods? Or can I just first do 333 (mod 17), it gives me 10. Then I could change all the 333's into 10s so it would be 10^10^10 and ...
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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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### Infinite Power Tower approximation: float error? [closed]

Desmos appears to plot it falsely using the $x^y = y$ definition, curving backwards. I've included a 50x exponent for comparison, which suggests no values flowing left in $x$-axis due to float error - ...
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### Comparing power towers of $2$s and $3s$

Let $x=[x_1,x_2,...,x_n]$ be a finite list of positive real numbers, and define $\tau x$ as the power tower formed by these numbers. The function $\tau$ can be recursively defined by the following two ...
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### How to compute $\,3^{3^{3^{\:\!\phantom{}^{.^{.^.}}}}}\!\!\!\!\bmod 46,$ for power tower height $2020$?

What is the remainder of $\,3^{3^{3^{\:\!\phantom{}^{.^{.^.}}}}}\!\!\!$ divided by $46$? The level of powers is $2020$. First there is no parenthesis so it means 3 power of 3 which is also power 3 ...
### How to solve $x^{x^{x^{x^{2010}}}} = 2010$
So I know that there is a difference between $(x^2)^3$ and $x^{2^3}$. But how do I use this knowledge to solve $$x^{x^{x^{x^{2010}}}} = 2010?$$