Tetration

I stumbled over this problem, thinking it to be just another tricky math problem:

\displaystyle \sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\cdot^{\cdot^{\cdot}}}}} = ?

Trying out by regular method of solving this problem, i.e removing one iteration in this infinite exponentiation, we get

\displaystyle (\sqrt{2})^x = x,

Now the above equation has two solutions which we see intuitively. x=2 as well as x=4 seem to satisfy the equation. This too doesn’t take me by surprise because in such situations one of them will be right and the other wrong because the LHS in the problem is a well defined real number and has to be unique. But, I recollected,

Any number greater than 1, raised to a power greater than 1 would always grow exponentially. Hence, an infinite exponentiation(tetration) of such number would yield infinity as the result.

I felt it to be quite intuitive! But here, I ended up with 2 solutions and both appear to be converging, while I expected it to be diverging. After a bit of googling I hit the right terms of Tetration and Infinite Power Tower Problem. I found out to my dismay, that it so happens,

Infinite tower power x^{x^{\cdot^{\cdot^{\cdot}}}} is continuous as a function, whose domain is the interval [e^{-e}, \sqrt[e]{e}] and whose range is [e^{-1},e].

I’m taken aback by the surprises math has in store to offer me. I thought I knew basic math and believed in some things as a thumb-rule which were so intuitive that they could not be denied. Its these mysteries, that drive us towards understanding things. Lesson learnt is,

However intuitive a thing might be, if you cannot prove it, you better not believe in it!

Jeremy Kun has a nice post, False proof 2=4 on validity of the solution and a link on detailed proof of the same.

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