Let $g(x) =\frac{2}{3} (x+\frac{1}{x^2})$, show that $g$ has a unique fixed point $r\in [1,2]$. what is it?
If $x_0∈[1,2]$, show that the functional iterates converge to $r$, $x_n→r$, and give the rate of convergence?
help me understand this i am pretty sure it's a contractive mapping theorem question but i am having a hard time understanding the procedure.
Contraction mapping theorem? Don't use bombs to kill ants. By definition of a fixed point, we have to solve $$ x = g(x) = \frac23 \left(x + \frac{1}{x^2} \right), $$ so we have to solve $$ x^3 = 2. $$ This equation a unique solution in $[1,2]$, namely $\sqrt[3]{2}$.