Re: x^(x^(x^(x^x^...)))....)))) = 2

On 2008-05-16, Sujit <sujit.gujar@xxxxxxxxx> wrote:
What is solution to
x^(x^(x^(x^x^...)))....)))) = 2?
( That is x raised to x raised to x ..infinite times... = 2)

As written it isn't very well defined. I'd formalise it a bit more by
defining the tetration operator ^^ in terms of repeated
exponentiation, and asking for a solution in x to lim(n->oo) x^^n = 2.


One immediate thing comes to mind is: it is same as x^(the whole
thing which is 2) =2 so x = sqrt 2

Yes, that is a solution to the equation.


but if we consider, x^(x^(x^(x^x^...)))....)))) = 4 still we get (if
apply same logic) x=sqrt 2

That isn't a solution if you define the problem in terms of limits.
The unique value of lim(n->oo) sqrt(2)^^n is 2. There is no solution
to lim(n->oo) x^^n = 4.

The idea of finding a limit of by looking at fixed points is not bad.
For a continuous function, the limit will be a fixed point: but the
converse is not necessarily true.


Surely if x=sqrt2 this series should blow up..and can't remain finite.....

If y < 2 then sqrt(2)^y < 2, so the series can't blow up.

One interesting question you might like to work on: how large can x be
without the series blowing up?


- Tim
.