Why does the recursion $x_{n+1} = f(x_n)$ with $f(x) = p^k\cdot\sqrt[k+1]{x}$ converge to $p^{k+1}$?
In addition, how can you write a function that has rational or integer fixed points? For example, you get $k^2$ as a fixed point of $f(x)=k\sqrt{x}$. Is there any known way to work back from an integer/rational number, and basically construct a function with said number as a fixed point?
Apparently, the solution is easier solved algebraically. Just let $x=p^{k}\cdot x^{\frac{1}{k+1}}$. Raising both sides by $k+1$ gives: $$\frac{x^{k+1}}{x}=p^{k(k+1)}$$ $$x^k=p^{k(k+1)}$$ $$x=p^{k+1}$$