Let $x_0=1,f:x\to x^5,x_{n+1}=x_n-\frac{F(x_n)}{F'(x_n)}$ with $F:x\to \frac{f(x)}{f'(x)}$. Does this series converge and with what rate of convergence?
This series converges to zero. And very fast, with just five iterations it will reach zero. Since we can write $x_{n+1}=x_n - \frac{f(x_n)f''(x_n)}{f'(x_n)^2}$ Plugging in we get $x_0=1,x_1=\frac45,x_2=\frac35,x_3=\frac25,x_4=\frac15,x_5=0$. So this convergence is super fast, but how can I prove its rate formally?