# Rate of convergence in newton raphson

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?

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