Linked Questions

5
votes
1answer
813 views

Values for which tetration to infinite heights (i.e., $x^{x^{x^{x^{.^{.^{.}}}}}}$) converges [duplicate]

I was reading Evaluating tetration to infinite heights (e.g., $2^{2^{2^{2^{.^{.^.}}}}}$). For what values of $x$ does tetration to infinite heights (i.e., $x^{x^{x^{x^{.^{.^{.}}}}}}$) converge?
4
votes
1answer
160 views

For what values does $x^{x^{x^{x^{.^{.^{.}}}}}}$ make sense [duplicate]

For which values of $x\ge 1$ does the expression $x^{x^{x^{x^{.^{.^{.}}}}}}$ make sense? To tackle this, define $f_1(x)=x$ and $f_{n+1}(x)=x^{f_n(x)}$ for $x \ge 1$ and $n\ge1$. a) Show that $f_{n+...
9
votes
0answers
258 views

Largest $x$ such that the power tower (tetration) $x^{x^{x^{x^{…}}}}$ converges? [duplicate]

Possible Duplicate: Infinite tetration, convergence radius Recently in this thread, Pseudo Proofs that are intuitively reasonable, I learned that $$\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{...}}}}...
5
votes
0answers
153 views

When does $x^{x^{x^{…}}}$ converge? [duplicate]

Provided that $x$ is positive, when does the following converge? $$x^{x^{x^{x^{...}}}}$$ So here is my work, but I don't really know if this is correct. Could anyone shed some light on this? I ...
0
votes
0answers
48 views

The Infinite Tetration of x? [duplicate]

The infinite tetration of ${x}$ basically means that we take a value and we continue to raise the value to its power forever. If that sounds confusing, it can be thought of as infinite repeated ...
-1
votes
1answer
25 views

Power tower(infinite tetration) [duplicate]

How to find the domain and range of $Y=x^{x^{x^{x^{x^{...}}}}}=x^Y$ I know how to differentiate the function. I don't know how to proceed further. We should prove that domain:$[1/(e^e),e^{(1/e)}]$ ...
75
votes
4answers
7k views

Are these solutions of $2 = x^{x^{x^{\:\cdot^{\:\cdot^{\:\cdot}}}}}$ correct?

Find $x$ in $$ \Large 2 = x^{x^{x^{\:\cdot^{\:\cdot^{\:\cdot}}}}}$$ A trick to solve this is to see that $$\large 2 = x^{x^{x^{\:\cdot^{\:\cdot^{\:\cdot}}}}} \quad\implies\quad 2 = x^{\Big(x^{x^{x^{...
0
votes
2answers
149 views

Solve X=sqrt(A)^sqrt(A)^sqrt(A)^…infinty? [duplicate]

If $X= \newcommand{\W}{\operatorname{W}}\sqrt{A}^{\sqrt{A}^{\sqrt{A}^{\sqrt{A}^{\sqrt{A}^{\sqrt{A}^{\sqrt{A}^{\sqrt{A}^{.{^{.^{\dots}}}}}}}}}}} $ then what is the value of $X^2-e^{1/X}$ ?
3
votes
2answers
421 views

Solving $4=2^{x^{x^{x^{x^{…}}}}} $ for $ x$

Is it possible to solve the following equal for $x$? $$4=2^{x^{x^{x^{...}}}}$$ I'm bit confused, how do you even simplify this equation, factoring?
0
votes
1answer
117 views

Evaluating $\sqrt2^{\sqrt2^{\sqrt2^{\:^{.^{\:^{.^{\: \infty}}}}}}}$ [duplicate]

The question is to evaluate $$\sqrt2^{\sqrt2^{\sqrt2^{\:^{.^{\:^{.^{\: \infty}}}}}}}$$ Let $$x^{x^{x^{x^{\;^{\cdots{^\infty}}}}}}=2$$ Then $x^2=2 \implies x=\sqrt2$ Now Let $$x^{x^{x^{\:^{.^{\:^{{. ...