Disclaimer: I do not have a maths degree, but is just one curious bloke. Be nice.
What do you call a function whose answer is repeatedly inserted into the function? For example $f(x) = 2x^2 +2$. If $x=1$, then $f(x)=4$. Plug in $x=4$, thus $f(x)=34$, and then plug $x=34$ to $f(x)$, and so on.
Given that property of the function, can anyone explain why this function:
$$f(x)=k\sqrt[2]{x}$$
converges to $k^2$ if you perform several iterations of the process? In general, what is this process/property called? Are there any other quirks like this? Thanks.
Fix $x$. Take $a_n=k\sqrt{k\sqrt{k\sqrt{...k\sqrt{x}}}} \quad (n\quad times)$ therefore we have $a_{n+1}=k\sqrt{a_n}$. For $k=0$ it's trivial so take $k\ne0$. Now define $b_n=\frac{a_n}{k^2}$ then:$$b_{n+1}=\frac{|k|}{k}\sqrt{b_n}$$ for $b_n$'s being real we must have $k>0$ so $b_{n+1}=\sqrt{b_n}$. If $b_1\ge 1$ so is $b_2$, $b_3$ and ... and the sequence is monotonically decreasing because $1\le b_{n+1}=\sqrt{b_n}\le b_n$ and since it has $1$ as a lower bound it's convergent ,say to $l\ge 1$. Therefore $l=\sqrt l$ which deduces $l=0$ or $l=1$ where $l=1$ is valid. Investigating $b_n$'s when $b_1\le 1$ is likewise. So $lim_{n\to\infty}b_n=1$ which implies $lim_{n\to\infty}a_n=k^2$