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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2
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2answers
106 views

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 ...
7
votes
1answer
351 views

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 ...
7
votes
1answer
235 views

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 ...
2
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0answers
75 views

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 ...
7
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0answers
313 views

Does $\mathrm{\int W(ln(x))dx}$ have a closed form?

This is follow up to this question which you will have to see for context: Is there a better solution for $$\mathrm{\int (a^t)^{(a^t)}dt= C+t+\frac1{ln(a)}\sum_{n=0}^\infty \frac{(-1)^n Q(n+1,-nt\,ln(...
0
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2answers
136 views

Solution to the Equation $x^x = 0$.

If I solve the Equation $x^x = 0$, then I get $x^x=0$: x ln⁡(x)= -∞ or ln⁡x $e^{\ln⁡(x)}$ =-∞ and if we take the Lambert W function on both sides we get ln⁡(x)=∞, And I put it on Wolfram Alpha then it ...
4
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1answer
419 views

Prove or disprove that the function is convex .

It seems we have : Define $\displaystyle f(x)=\sum_{k=1}^{2n}x^{k^2}$ where $n\geq 1$ a natural number and $-1\leq x\leq 1$ Claim : $f''(x)\geq 0$ My attempt : The case $n=1$ is trivial . So I have ...
4
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2answers
481 views

Correct my proof about : $x^{x^{x^{x^{x^{x}}}}}$ is convex for $0.25<x<0.27$

Let $0.25<x<0.27$ then define : $$f(x)=x^{x^{x^{x^{x^{x}}}}}$$ Claim: The second derivative of $f(x)$ is strictly positive . Proof : Let $0.25<x<0.27$ then define : $$g(x)=x^{x^{x^{x}}}$$ ...
5
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1answer
323 views

Is there a better solution for $\mathrm{\int (a^t)^{(a^t)}dt= C+t+\frac1{\,ln(a)}\sum_{n=0}^\infty \frac{(-1)^n Q(n+1,-nt\ln(a))}{n^{n+1}}\,dt}$?

I know there exist functions like this one for simplifying tetration based sums. There may be a way to simplify this type of sum at least using a lesser known and widely accepted functions. Here are ...
4
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1answer
140 views

How to evaluate the finite power tower $\tan(1°)^{\tan(2°)^{\tan(3°)^{\cdot^{\cdot^{\cdot^{\tan(44°)^{\tan(45°)}}}}}}}$

Consider the following finite power tower: $$\Large \tan(1°)^{\tan(2°)^{\tan(3°)^{\cdot^{\cdot^{\cdot^{\tan(44°)^{\tan(45°)}}}}}}}$$ I'm wondering if there is a way to solve this that doesn't rely on ...
6
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1answer
147 views

Is the infinite product $\prod_{i=0}^{\infty}(1+\frac{1}{2^{3^i}})$ transcendental?

Is the following number algebraic or transcendental? $$P:=\prod_{i=0}^{\infty}\left(1+\frac{1}{2^{3^i}}\right)$$ We could also define it as follows: let A be the set of natural numbers which contain ...
1
vote
1answer
192 views

On $\int \sqrt[x]x$dx. Solution found for $eW\left(\frac1e\right)\le x\le e$.

Notice: Note that there is hope for a more general answer because an answer for the area under the infinite tetration/power tower curve was almost certainly found. It uses the nice OEIS A008405 find. ...
14
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1answer
508 views

About the inequality : $x^{x^{x^{x^{x^x}}}}\geq x^{x^{x^{((e-2)(1+e))x\left(1+\sqrt{x}\left((\sqrt{x})^3-1\right)\right)}}}\geq x^{x^{\frac{16}{27}}}$

This inequality is due to user RiverLi : Let $0<x\leq 1$ then we have : $$x^{x^{x^{x^{x^x}}}}\geq x^{x^{\frac{16}{27}}} \geq 0.5x^2+0.5$$ I propose another one wich states : Let $0<x\leq 1$ ...
3
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0answers
121 views

Stronger statement : $^{6}x\geq e^{\left(\frac{\ln^{2}\left(x+1\right)}{\ln\left(2\right)}-\ln\left(2\right)\right)}\geq \left(x^{2}+1\right)0.5$

Let $x>0$ then prove or disprove that : $$x^{x^{x^{x^{x^{x}}}}}\geq e^{\left(\frac{\ln^{2}\left(x+1\right)}{\ln\left(2\right)}-\ln\left(2\right)\right)}\geq \left(x^{2}+1\right)0.5$$ My attempt : ...
17
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2answers
717 views

Integral over domain of infinite tetration of x over extended domain from 0 to $\sqrt[e]e$. Possible $\int_{e^{-e}}^{e^\frac1e} x^{x^{…}}dx$ solution.

I have been trying to find an interesting constant over the domain of the infinite tetration of x and have just almost figured out the area with a non integral infinite sum representation. Just one ...
19
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3answers
665 views

Prove or disprove that the function $f(x)=x^{x^{x^{x}}}$ is convex on $(0,1)$

Let $0<x<1$ and $f(x)=x^{x^{x^{x}}}$ then we have : Claim : $$f''(x)\geq 0$$ My attempt as a sketch of partial proof : We introduce the function ($0<a<1$): $$g(x)=x^{x^{a^{a}}}$$ Second ...
7
votes
2answers
473 views

Area under $x^{-x}$ over its real domain. What is another non-integral form of $\int_{\Bbb R^+}x^{-x}dx$?

A few years ago, I got interested in an apparently hard integration problem which had me fascinated. This was the integral of a Sophomore Dream like integral except with the bounds over the real ...
7
votes
1answer
241 views

Limits of negative-power tower

Consider the following: $$n \uparrow -\Bigl((n+1) \uparrow -\bigl((n+2) \uparrow \cdots \uparrow -m \bigr)\Bigr)= n^{{{-(n+1)}^{-(n+2)}}^{\cdots^{-m}}}$$ It doesn't converge for $m \to \infty$, but ...
2
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1answer
63 views

If $\displaystyle x^{x^9}=\sqrt{3^{\sqrt{3}}}$ and $\displaystyle y=x^{\left(\frac{1}{y^{y^x}}\right)}$, determinate the value of $y^{3x}$.

If $\displaystyle x^{x^9}=\sqrt{3^{\sqrt{3}}}$ and $\displaystyle y=x^{\left(\frac{1}{y^{y^x}}\right)}$, determinate the value of $y^{3x}$. My try It is easy to see that if we raise the first equation ...
2
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0answers
74 views

Is the infinite tower power $e^{e^{e^{e...}}}$ defined?

Here's what happens when I try to evaluate the infinite tower power, $e^{e^{e^{e...}}}$: $$x=e^{e^{e^{e...}}}$$ $$x=e^x$$ $$\ln x =x$$ $$\ln x=e^x$$ No Real solution. But come on! All those piled up e'...
1
vote
1answer
86 views

did I make a mistake? when trying to find the derivative of $x^{x^{x^{x^{x^{\dots}}}}}$

did I make a mistake? when trying to find the derivative of $f(x)=x^{x^{x^{x^{x^{\dots}}}}}$ the first thing I did was to change the $f(x)$ to this $$y=x^y$$ I took the derivative of both $y$ and $x$ ...
0
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0answers
41 views

Representing a Number as the Sum of Powers of Form $k^k$.

So I was wondering if it would be useful to instead of writing a number in base $2$ or $3$, we use functions in general as bases. So like writing it as the sum of squares or other increasing functions....
3
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0answers
45 views

Solution Verification - Integral of infinite power tower

$$\int x^{x^{\cdot^{\cdot^\cdot}}}dx=\int -\frac{W(-\ln(x))}{\ln(x)}dx$$ Let $u=-\frac{W(-\ln(x))}{\ln(x)}$, $\frac{du}{dx}=\frac{u^2}{(1-\ln(u))u^{\frac{1}{u}}}\Rightarrow dx=\frac{(1-\ln(u))u^{\frac{...
7
votes
1answer
345 views

Find a value of $\;\lim\limits_{n\rightarrow\infty}n\left ( e- e^{\frac{1}{e}}\uparrow\uparrow n \right )$

Find a value of $$\lim_{n\rightarrow\infty}n\left ( e- e^{\frac{1}{e}}\uparrow\uparrow n \right )$$ For your information$,\quad\uparrow\uparrow$ is a tetration defined as $$a\uparrow\uparrow n:=\...
2
votes
2answers
119 views

Integrating an Infinite exponent tower

$$ \int_0^1 x^{2^{x^{2^{x^{\ldots}}}}} ~~ dx = ~~?$$                                                                                                                                           What I'...
0
votes
6answers
160 views

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 ...
3
votes
1answer
207 views

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, ...
3
votes
1answer
73 views

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 & \;\...
1
vote
1answer
95 views

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 ...
1
vote
1answer
63 views

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 ...
2
votes
1answer
88 views

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 ...
-1
votes
1answer
84 views

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 - ...
4
votes
2answers
100 views

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 ...
1
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0answers
21 views

inequality $\Re\Bigg(\Bigg(1-\frac{W(-\ln(2a))}{\ln(2a)}\Bigg)^{1-\frac{1}{a}}+\Bigg(1-\frac{W(-\ln(2b))}{\ln(2b)}\Bigg)^{1-\frac{1}{b}}\Bigg)<1$

Problem inspired by Vasile Cirtoaje : Let $0<b<0.5<a<1$ such that $a+b=1$ then we have : $$\Re\Bigg(\Bigg(1-\frac{W(-\ln(2a))}{\ln(2a)}\Bigg)^{1-\frac{1}{a}}+\Bigg(1-\frac{W(-\ln(2b))}{\...
0
votes
2answers
94 views

how many $\sqrt{2}^{\sqrt{2}^{\sqrt{2}\ldots }}$

Use $y=x^{\frac{1}{x}}$ graph and think the following calculate. $$\sqrt{2}^{\sqrt{2}^{\sqrt{2}\ldots }}$$ I want to know how many $\sqrt{2}^{\sqrt{2}^{\sqrt{2}\ldots }}$ will be. When $\sqrt{2}^{\...
4
votes
1answer
183 views

Is it possible to express $\int_{0}^{1-\epsilon}\left(\sqrt{1-x^2}^{\sqrt{1-x^2}^{\cdots}}\right) dx$ in elementary functions?

let $\epsilon >0$, I tried to evaluate $\int_{0}^{1}\left(\sqrt{1-x^2}^{\sqrt{1-x^2}^{\cdots}}\right) dx$ , using the fact $x= \cos t$ yield to have integrand using $\sin $ function seems is not ...
1
vote
0answers
78 views

"Shooting Room" ratio when selection sizes grow tetrationally

A version of the Shooting Room "paradox" involves selecting disjoint sets in the plane, first selecting a set of area $1$, then a set of area $r$, then of $r^2$, etc.--i.e. geometric growth--...
0
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1answer
115 views

Does the infinite power tower $x^{x^{x^{x^{x^{x^{x^{.{^{.^{.}}}}}}}}}}$ converge for $x < e^{-e}$?

As per the answer at https://math.stackexchange.com/a/573040/23890, the infinite power tower $x^{x^{x^{x^{x^{x^{x^{.{^{.^{.}}}}}}}}}}$ converges if and only if $ x \in [e^{-e}, e^\frac{1}{e} ] $. Is $...
1
vote
1answer
63 views

$i^i\in\mathbb R$

I knew that $^2i=i^i\in\mathbb R$ so I tried to find another number $n\in\mathbb N$ like that: $^ni\in\mathbb R$, so, there is no such number all the way up to a million, can we prove that two is the ...
0
votes
1answer
66 views

Why it were conjectured that $e^{e^{^e{^{79}}}}$ is not an integer only for $n=79$ ? any non trivial characterization?

I'm confused that why exactly and what is the reason to conjecture that $e^{e^{^e{^{79}}}}$ is not integer , why not for example with $n=87$ or any other prime $p$ ?Is this number special ? or is ...
12
votes
2answers
359 views

Convergence of probabilistic power tower $e^{\pm e^{\pm e^{...}}}$

While pondering over this question, I came across another interesting one. I am familiar with infinite tetration and its convergence over the reals. Nevertheless, when I saw this power tower, I couldn'...
1
vote
0answers
32 views

Almost integer with nested radicals and power tower .

playing with power tower and nested radicals I get : Prove that Let $a_1=\sqrt{2}$ ,$a_2=\sqrt{2}^{\sqrt{2}}$,$a_3=\sqrt{2}^{\sqrt{2}^{\sqrt{2}}}$,$a_4=\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}}}...
2
votes
0answers
26 views

Comparison of power towers with different bases

If I define $T(c, k, r) = c^{c^{\cdot^{\cdot^{\cdot^{c^r}}}}}$ with $(k-1)$ $c$'s in the tower. I want to understand the asymptotic behaviour of towers with different values of $c$. I am specifically ...
1
vote
2answers
64 views

Find the last digits of $a_{2009}$, and of $b_{2009}$.

Define the sequences $a_1, a_2,...$ and $*b_1, b_2,...*$ by $a_1 = b_1 = 7$ and $$a_{n+1} = {a_n}^7, \\ b_{n+1} = 7^{b_n}$$ for $n\ge 1$. Find the last digits of $a_{2009}$, and of $b_{2009}$. What ...
4
votes
1answer
178 views

Alternating power sequence

I quite randomly stumbled upon the following phenomenon: Let $ f:\mathbb R^+\to\mathbb R^+, x\mapsto x^{-2^{3^{-4^{\cdot^{\cdot^{\cdot}}}}}} $, then in the interval of $[1,20)$ the plot of $f$ looks ...
5
votes
0answers
111 views

Complexity of repeatedly applying Euler's totient

When performing modular power towers e.g. $a_0^{a_1^{a_2^{.^{.^.}}}}\bmod n$, Euler's totient theorem and it's generalization reduces the problem to computing $$n,\varphi(n),\varphi(\varphi(n)),\...
-2
votes
2answers
170 views

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 ...
2
votes
3answers
123 views

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 $$x^{x^{x^{x^{2010}}}} = 2010?$$
3
votes
1answer
78 views

Simple power tower $x^{1-x+x^{1-x+x^{1-x+x^{1-x+\cdots}}}}=x$

I was not clear in my last post : Let $0<x$ a real number then we have : $$x^{1-x+x^{1-x+x^{1-x+x^{1-x+\cdots}}}}=x$$ I take the logarithm of both side we get : $$\ln(x)({1-x+x^{1-x+x^{1-x+x^...
1
vote
0answers
85 views

Sum of power tower which tends to $1$

Inspired by an inequality of Vasile Cirtoaje, I have this : Let $0<x<1$ then : $$x^{2(1-x)}+(1-x)^{2x}\leq 1$$ Let $0<x<1$:$$f(x)=x^{1-x+x^{1-x+x^{2 (1-x)}}}$$ Then : ...

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