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?

Related is Problem regarding convergence of second order

  • $\begingroup$ It seems like you have $F'(x_n)$ incorrectly there. $\endgroup$ – Chee Han Jan 31 '18 at 18:01
  • $\begingroup$ yes you are right, then $F'(x)=- \frac{f(x)}{f''(x)^2}+1$ $\endgroup$ – Andi Jan 31 '18 at 18:11
  • $\begingroup$ @CheeHan I am unable to solve this problem, could you show me how to do this? $\endgroup$ – Andi Jan 31 '18 at 18:22

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