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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Find the value of $\frac{1}{2}^{\frac{1}{3}^{\frac{1}{4}^{\cdots \infty}}}$? [duplicate]
Greetings with utmost respect, everyone!
Today, I found a fascinating math question online. I seem to be stuck while solving it, however. I did not find any relevant solution to the problem yet. ...
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Solve for $m(t)$ in the integral transform $\int_0^1(1-t^n) m(t) dt=\frac{(n+1)^n}{n^{n-1}} $ for $n>0$.
Background
(You can skip this part, but maybe you find it interesting.)
Is $ \displaystyle f_1(x,v) = \sum_{n=0}^{\infty} \frac{x^n}{(n!)^v } > 0 $ for all real $x$ and $0<v<1$ ?
Lets start ...
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Numbers such that $ \sqrt{ {a_{1}}^{a_{2}^{...^{a_{n}}}}} = a_{1} a_{2} \dots a_{n}$
I wonder whether there are any resources on the equation:
$$\require{\MnSymbol} \require{\mathdots} \sqrt{ {a_{1}}^{a_{2}^{...^{a_{n}}}}} = a_{1} a_{2} \dots a_{n}. \tag{1}$$
Here, $a_{1}, a_{2}, \...
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Convergence of Variable Base Power Tower $\frac{1}{2}^{\frac{1}{3}^{\frac{1}{4}^{\ldots}}}$
I’m curious if there is a heuristic method that can be used to solve what appears to be an elementary power tower where every increasing power is decreasing by a half integer. Numerical methods ...
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Infinite power tower for $x > e^{1/e}$
For $x>0$, let $\tau_0 = 1$ and $\tau_{n+1} = x^{\tau_n}$. The infinite power tower of $x$ is then $\tau = \tau(x) = \lim_{n\to \infty} \tau_n$. It is well known that $\tau$ exists and is finite ...
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Solve the tower equation: $\displaystyle x^{x^{x^{x+1}+x+1}}=2$
We are struggling to solve a crazy looking equation below:
$$\displaystyle x^{x^{x^{x+1}+x+1}}=2$$
Approximate numerical values are unfortunately not a solution. Wolfram Alpha offers only the ...
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1
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Can I find $x^{x^{-1}}$ given $x^{-17^{ 17^{-x}}}=17$?
Has This problem a solution? I tried for many ways, even Mathematica. I think it has a typo:
$$x^{-17^{ 17^{-x}}}=17$$
Find: $$x^{x^{-1}}$$
How can I explain it has no solution.
Kind regards.
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How to solve the ODE $(y(x))^{((y'(x))^{((y''(x))^{((y'''(x))^{\cdot^{\cdot^{\cdot}}})})})}=f(x)$?
Introduction
Why This Question?
I have often asked myself what a solution to the Ordinary Differential Equation (ODE) would be and when I recently saw this ODE again, tried it again and failed again, ...
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If $\lim_{n\to \infty} a_n = 4$, can we say $\lim_{n\to \infty} a_{n-2} = 4$?
I was doing this exponential tower equation:
$$2^{x^{2^{x^{2^{...}}}}} = 4$$
$$\text {(each new exponent is the power of the last exponent)}$$
The popular method is to break the tower at the first ...
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1
answer
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Closed form of $\begin{align}\int_{0}^{\infty}x^{1-x^{2-x^{3-...}}}dx\end{align}$.
I want to find the closed form of $\begin{align}\int_{0}^{\infty}x^{1-x^{2-x^{3-...}}}dx\end{align}$.
Quick disclaimer: I have no reason to believe one actually exists
Using Desmos, the closest I have ...
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2
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How to solve $x^{y^z}=z$
Initially I isolated the y in $x^y=y$, but I just wanted to expand the infinite power tower to two letters in the tower, but I can't solve for z in the equation $x^{y^z}=z$. I tried to use Lambert W ...
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Operation that Turns Powers into Products Like How Logarithms Turn Products into Sums
$$
\newcommand{\pow}{\mathop{\vcenter{\huge{\text{E}}}}\limits}
$$
The $\sum$ operator can be defined recursively as
$$
\sum_{i = a}^b f(i) = f(a) + \sum_{i = a + 1}^b f(i).
$$
Likewise, the $\prod$ ...
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1
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Power tower modulus
I'm doing a programming challenge and I can't wrap my head around of finding an Euler totient when the modulus and the base aren't co-primes.
So I have:
$$4 ^ {4 ^ 4}(mod\;10)$$
I understand that 4 ...
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2
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i raised to itself
Last night I learned the amazing fact that $i^i=e^{-\pi/2}$
So I started computing other powers and would like to get confirmation that:
$i^{i^i}=e^{-i\pi/2}$
$i^{i^{i^{i}}}=e^{\pi/2}$
$i^{i^{i^{i^{i}}...
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1
answer
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Power rules: $x ^ {(y^z)}$
I know about $(x^y)^z$ = $x^{yz}$ but what about $x^{(y^z)}$? Are there any rules for this?
Let's consider the power of power rule: $(x^y)^z$ = $x^{yz}$:
$(2^3)^4$ = $8^4$ = 4096
$2^{3\times 4}$ = $2^...
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3
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$x^{x^2} + x^{x^8} =?$ Given $x^{x^4} = 4$
If x is any complex number such that $x^{x^4} = 4$ , then find all the possible values of :
$x^{x^2} + x^{x^8}$
First, I used laws of exponents to give $18$ as answer. However , I realised that I've ...
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Find $f$ such that $f(f(x))=\log (x)$
Which is the solution for the following expression $$f(f(x))=\log (x)?$$
In other words, which function composed with itself matches with logarithm? With $x\in (0,\infty )$.
The breakdown of ...
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Stack the digits of number
$0^0$ is defined as $1$ in this question.
1.I have a natural number $N$, $N=\overline{a_1a_2a_3\cdots a_n}$, where $\overline{a_1a_2a_3\cdots a_n}$ denotes the number in decimal representation.
2.If $...
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Digits tower power iterate
Stack the digits of a natural number into a power tower, iterate until only one digit remains.
Does this iteration always terminate for any positive integer?
Additionally specify $0^n = 0$, even when $...
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1
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Correct my sketch of proof about the convexity of the "natural" power tower on $[1,\infty)$
Hi I want to show the following fact :
Problem :
Let $x\geq 1$ and $n\geq 1$ a natural number and define:
$$f(x)={}^{2n}x=\underbrace{x^{x^{⋰^{x}}}}_{2n\text { times}}$$
Then we have :
$$f''(x)\ge 0$$
...
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1
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$f(n)=$, for even integer’s $n$, $\lim_{n\to\infty} \int_{0}^{\infty} \frac{e^x}{^nx} dx$
$f(n)=$, for even integer’s $n$, $$\lim_{n\to\infty} \int_{0}^{\infty} \frac{e^x}{^nx} dx$$
$^n x$ is the tetrative function or ‘power tower’, which when $n=\infty$ can also be written in terms of y, ...
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Is each of $\int_0^\infty\frac{dx}{x^x},\int_0^\infty\frac{dx}{x^{x^{x^x}}},\int_0^\infty\frac{dx}{x^{x^{x^{x^{x^x}}}}},\cdots$ less than $2$?
A few years ago I asked about the inequality Prove that $\int_0^\infty\frac1{x^x}\, dx<2$. As I came back to revisit it, I found that each of the following tetration integrals $$\int_0^\infty\frac{...
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Examples of closed forms of integrals with a power tower argument using W-Lambert function.
Here is a closed form of an integral that looks like:
Integral form(s) of a general tetration/power tower integral solution: $$\sum\limits_{n=0}^\infty \frac{(pn+q)^{rn+s}Γ(an+b,cn+d)}{Γ(An+B)}$$
...
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Evaluating negative infinite tetration: $\lim\limits_{n\to-\infty}\,^n x=\lim_{n\to\infty}\underbrace{\log_x\left(…\log_x(0)\right)}_n=\,^{-\infty} x$
One can learn that for a power tower of height $n$:
$$y=n\big\{x^{x^{x^…}}=\,^nx\implies \log_x(y)=\boxed{\log_x(\,^nx)=\,^{n-1}x}$$
giving a recursive relation. One might see that $n<-2$ cases are ...
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Could someone explain why $\sum_{\substack{a_1,\ldots,a_n\in\mathbb{N}_0\\a_1+\cdots+a_n=n}}\frac{n!}{a_1!\cdots a_n!}=n^n$?
I want to use this equality but I have no idea why it holds. Sure I can probably prove it via induction but it looks rather fiddly. (Let $n$ be a positive integer.)
$$\sum_{\substack{a_1,\ldots,a_n\in\...
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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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0
answers
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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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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 $$\mathrm{\int (a^t)^{(a^t)}dt= C+t+\frac1{ln(a)}\sum_{n=0}^\infty \frac{(-1)^n Q(n+1,-nt\,ln(...
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Solution to the Equation $x^x = 0$.
If I solve the Equation $x^x = 0$, then I get $x^x=0$:
x ln(x)= -∞
or
lnx $e^{\ln(x)}$ =-∞
and if we take the Lambert W function on both sides we get
ln(x)=∞,
And I put it on Wolfram Alpha then it ...
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Prove or disprove that the function is convex .
It seems we have :
Define $\displaystyle f(x)=\sum_{k=1}^{2n}x^{k^2}$ where $n\geq 1$ a natural number and $-1\leq x\leq 1$
Claim :
$f''(x)\geq 0$
My attempt :
The case $n=1$ is trivial . So I have ...
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Correct my proof about : $x^{x^{x^{x^{x^{x}}}}}$ is convex for $0.25<x<0.27$
Let $0.25<x<0.27$ then define :
$$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}}}$$
...
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Is there a better solution for $\mathrm{\int (a^t)^{(a^t)}dt= C+t+\frac1{\,ln(a)}\sum_{n=0}^\infty \frac{(-1)^n Q(n+1,-nt\ln(a))}{n^{n+1}}\,dt}$?
I know there exist functions like this one for simplifying tetration based sums. There may be a way to simplify this type of sum at least using a lesser known and widely accepted functions. Here are ...
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How to evaluate the finite power tower $\tan(1°)^{\tan(2°)^{\tan(3°)^{\cdot^{\cdot^{\cdot^{\tan(44°)^{\tan(45°)}}}}}}}$
Consider the following finite power tower:
$$\Large \tan(1°)^{\tan(2°)^{\tan(3°)^{\cdot^{\cdot^{\cdot^{\tan(44°)^{\tan(45°)}}}}}}}$$
I'm wondering if there is a way to solve this that doesn't rely on ...
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Is the infinite product $\prod_{i=0}^{\infty}(1+\frac{1}{2^{3^i}})$ transcendental?
Is the following number algebraic or transcendental? $$P:=\prod_{i=0}^{\infty}\left(1+\frac{1}{2^{3^i}}\right)$$
We could also define it as follows: let A be the set of natural numbers which contain ...
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On $\int \sqrt[x]x$dx. Solution found for $eW\left(\frac1e\right)\le x\le e$.
Notice:
Note that there is hope for a more general answer because an answer for the area under the infinite tetration/power tower curve was almost certainly found. It uses the nice OEIS A008405 find. ...
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About the inequality : $x^{x^{x^{x^{x^x}}}}\geq x^{x^{x^{((e-2)(1+e))x\left(1+\sqrt{x}\left((\sqrt{x})^3-1\right)\right)}}}\geq x^{x^{\frac{16}{27}}}$
This inequality is due to user RiverLi :
Let $0<x\leq 1$ then we have :
$$x^{x^{x^{x^{x^x}}}}\geq x^{x^{\frac{16}{27}}} \geq 0.5x^2+0.5$$
I propose another one wich states :
Let $0<x\leq 1$ ...
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Stronger statement : $^{6}x\geq e^{\left(\frac{\ln^{2}\left(x+1\right)}{\ln\left(2\right)}-\ln\left(2\right)\right)}\geq \left(x^{2}+1\right)0.5$
Let $x>0$ then prove or disprove that :
$$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$$
My attempt :
...
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2
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Integral over domain of infinite tetration of x over extended domain from 0 to $\sqrt[e]e$. Possible $\int_{e^{-e}}^{e^\frac1e} x^{x^{…}}dx$ solution.
I have been trying to find an interesting constant over the domain of the infinite tetration of x and have just almost figured out the area with a non integral infinite sum representation. Just one ...
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3
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Prove or disprove that the function $f(x)=x^{x^{x^{x}}}$ is convex on $(0,1)$
Let $0<x<1$ and $f(x)=x^{x^{x^{x}}}$ then we have :
Claim :
$$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}}}$$
Second ...
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votes
2
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Area under $x^{-x}$ over its real domain. What is another non-integral form of $\int_{\Bbb R^+}x^{-x}dx$?
A few years ago, I got interested in an apparently hard integration problem which had me fascinated. This was the integral of a Sophomore Dream like integral except with the bounds over the real ...
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9
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1
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Limits of negative-power tower
Consider the following:
$$n \uparrow -\Bigl((n+1) \uparrow -\bigl((n+2) \uparrow \cdots \uparrow -m \bigr)\Bigr)= n^{{{-(n+1)}^{-(n+2)}}^{\cdots^{-m}}}$$
It doesn't converge for $m \to \infty$, but ...
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If $\displaystyle x^{x^9}=\sqrt{3^{\sqrt{3}}}$ and $\displaystyle y=x^{\left(\frac{1}{y^{y^x}}\right)}$, determinate the value of $y^{3x}$.
If $\displaystyle x^{x^9}=\sqrt{3^{\sqrt{3}}}$ and $\displaystyle y=x^{\left(\frac{1}{y^{y^x}}\right)}$, determinate the value of $y^{3x}$.
My try
It is easy to see that if we raise the first equation ...
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1
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did I make a mistake? when trying to find the derivative of $x^{x^{x^{x^{x^{\dots}}}}}$
did I make a mistake? when trying to find the derivative of $f(x)=x^{x^{x^{x^{x^{\dots}}}}}$
the first thing I did was to change the $f(x)$ to this $$y=x^y$$
I took the derivative of both $y$ and $x$ ...
user913631
0
votes
0
answers
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Representing a Number as the Sum of Powers of Form $k^k$.
So I was wondering if it would be useful to instead of writing a number in base $2$ or $3$, we use functions in general as bases. So like writing it as the sum of squares or other increasing functions....
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0
answers
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Solution Verification - Integral of infinite power tower
$$\int x^{x^{\cdot^{\cdot^\cdot}}}dx=\int -\frac{W(-\ln(x))}{\ln(x)}dx$$ Let $u=-\frac{W(-\ln(x))}{\ln(x)}$, $\frac{du}{dx}=\frac{u^2}{(1-\ln(u))u^{\frac{1}{u}}}\Rightarrow dx=\frac{(1-\ln(u))u^{\frac{...
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1
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Find a value of $\;\lim\limits_{n\rightarrow\infty}n\left ( e- e^{\frac{1}{e}}\uparrow\uparrow n \right )$
Find a value of
$$\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:=\...
user822157
2
votes
2
answers
261
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Integrating an Infinite exponent tower
$$ \int_0^1 x^{2^{x^{2^{x^{\ldots}}}}} ~~ dx = ~~?$$
What I'...
0
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6
answers
176
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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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