On internet I found some recreational problems as $3=\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x+...}}}}$ $\frac{1}{2}=\frac{1}{x+\frac{1}{x+...}}$ $2=x^{x^{x^{...}}}$ And the trick to solve them was just to reuse the equation itself inside the equation, so for example obtaining $3=\sqrt{x+3}$ in the first example above, and so on. But this "trick" reminded me of the well known pseudoparadox we could obtain manipulating divergent series as for example the Bolzano's $1-1+1-1+1-1+...$ that setting $S=1-1+1-1+1-1+...$ become $S=1-1+1-1+1-1+...=1-(1-1+1-1+1-1+...)=1-S$ and therefore $1-1+1-1+1-1+...=S=1/2$. Obviously the error is in assuming that this series has a precise value, naming it $S$.
We obtain other oddities by generalizing the equations above so that $a=\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x+...}}}}$ would imply $x=a(a-1)$, $a=\frac{1}{x+\frac{1}{x+...}}$ would imply $x=\frac{1}{a}-a$ and $a=x^{x^{x^{...}}}$ would imply $x=a^{\frac{1}{a}}$, that would be quite strange identities for lots of values of $a$.
Then I feel the need to show that the equations above make sense. I think the most obvious way to formalize them was to define them as limits, exempli gratia $\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x+...}}}}=\lim_{n\rightarrow \infty}\underbrace { \sqrt{x+\sqrt{x+\sqrt{x+...+\sqrt{x}}}} }_{n \text{ times} }$
Then, trying to generalize the problem, we want to study the convergence of the sequence of "recursive" function $F_0(x)=x \\ F_n(x)=f(x,F_{n-1}(x))$ For example, we obtain the example at the beginning by respectively setting $f(x,y)=\sqrt{x+{y}}$,$f(x,y)=\frac{1}{x+y}$ and $f(x,y)=x^y$.
In case we manage to prove the convergence, naming $l$ the limit of the sequence, we would have that $f(x,l)=l$.
I've tried to study the sequence $x,x^x,x^{x^x},x^{x^{x^{x}}},...$, for $x \in (0,1)$; after noticing that $\underbrace {x^{x^{x^{x}}} }_{2n \text{ times} }\leq \underbrace {x^{x^{x^{x}}} }_{2n+1 \text{ times} }$ and $\underbrace {x^{x^{x^{x}}} }_{2n+1 \text{ times} }\geq \underbrace {x^{x^{x^{x}}} }_{2n+2 \text{ times} }$, I suspected that for very small value, this succession does not converge, somehow oscillating... but I was not able to prove it.
My questions are:
- Is there some related theory about "infinite equations" as discussed here (that is, if my formalization is correct, the convergence of $F_n(x)=f(x,F_{n-1}(x))$)?
- What about the special case of $2=x^{x^{x^{...}}}$? Is there a way to prove or disprove the convergence for $x\in (0,1)$?