Power Tower

Calculus Level 2

$\huge a^{a^{a^{\cdot^{\cdot^\cdot}}}}=2$

How many positive real solutions $a$ does the above equation have?

Clarification : The value of the infinite power tower $a^{a^{a^{.^{.^.}}}}$ is defined as the limit of the sequence $x_0=a, x_{n+1}=a^{x_n}.$

Bonus : Illustrate your solution with a Cobweb Plot .

Before you write a solution, consider this problem .

3 Infinitely many 2 1 0

1 solution

Sagar Shah
Feb 12, 2016

Here a^2 = 2 as a^a^a^a... is equal to 2.

=> a = √2

Hence, there is 1 solution.

Your answer is correct but the solution is incomplete. With the same reasoning, you would find that the solution of $a^{a^{a^{...}}}=4$ is $a=\sqrt[4]{4}=\sqrt{2}$ as well. But the infinite power tower with $a=\sqrt{2}$ cannot attain both of the values 2 and 4. Which one is it, and why?

Otto Bretscher - 5 years, 4 months ago

Otto, you already proved this... Let $x_{0} = \sqrt{2}, x_{n+1} = \sqrt{2}^{x_{n}}, \forall n \in \mathbb{N}$ . This sequence is bounded for 2 (it can be proved by induction) and strictly increasing... then it has a limit x . Now, $f(x) = \sqrt{2}^{x}$ is a continuous function, so the limit x has to fulfill $x = \sqrt{2}^x$ , this equation has two solutions but only x = 2 is factible, and we have finished. Am I wrong?

Guillermo Templado - 5 years, 4 months ago

Yes, exactly! I'm posting this for people who might not have read our discussion here

Otto Bretscher - 5 years, 4 months ago

@Otto Bretscher – Can you please post your solution there?? So that people might understand their mistake since I would be deleting my solution...

Rishabh Jain - 5 years, 4 months ago

@Rishabh Jain – @Guillermo Templado has posted a fine solution... does it suffice? (I'm at work now, but I will post a more detailed solution in the evening if you wish.) It would be great if somebody could generate a Cobweb diagram... I feel that those explain the dynamics better than any write-up.

Otto Bretscher - 5 years, 4 months ago

@Otto Bretscher – I'm saying about this question..... Since many people who have got that problem right might think they are right( but actually they are not) and after I'll delete my wrong solution( on which your fruitful discussion took place) ... People will report that problem and ask for why is previous amswer not correct .. That's why I'm requesting you to post your solution on that problem..

Rishabh Jain - 5 years, 4 months ago

@Rishabh Jain – I cannot post a solution there since I did not get the "right" answer (which is really the wrong answer). ;) I hope that the administrators will get to my report soon.

Otto Bretscher - 5 years, 4 months ago

@Otto Bretscher – Right now that problem is Level 2 ... But as soon as your report gets resolved that problem gonna shoot to at least level 4 since not many people would would have realised that that simple problem isn't so simple.... :-}

Rishabh Jain - 5 years, 4 months ago

@Rishabh Jain – I have now written a solution to that "Exponentials" problem... please let me know whether it is sufficiently detailed to be understandable..

Otto Bretscher - 5 years, 4 months ago

@Otto Bretscher – Ya... Thanks :)... It is wonderful....

Rishabh Jain - 5 years, 4 months ago

@Otto Bretscher – Oh now I see the modified comment.... Perfect... Thanks..

Rishabh Jain - 5 years, 4 months ago

Is sqrt(2) is real?

Irvine Dwicahya - 5 years, 4 months ago

Yes of course

Otto Bretscher - 5 years, 4 months ago

I think that's the correct option is "2", because $\pm \sqrt{2}$ other solution is minus.

chris wild - 2 years, 12 months ago