This is a question from D Kincaid & W Cheney, Numerical Analysis (3ed), Brooks-Cole 2002;
Find the condition on $\alpha$ to ensure that the iteration will converge linearly to a zero of $f$ if started near the zero.
This question is in the Newton's Method Section, I tried to get a solution using ideas from that section, but I am not sure,
Say $f(r)=0$ and and $x_{n}-r=e_{n}$ then using $x_{n+1}=x_{n}-\alpha f(x_{n})$. we get $e_{n+1}=e_{n}-\alpha f(x_{n})$.
Using Taylor's thm we have 0=f(r)=f(x_{n}-r)=f(x_{n})-e_{n}f'(\eta_{x}) for some $\eta_{x}$ between $r$ and $x_{n}$.
Using this in above we get e_{n+1}=e_{n}-\alpha f(x_{n})=e_{n}-\alpha e_{n} f'(\eta_{x}) \approx e_{n}(1-\alpha f'(r))
To have linear convergence we need (1-\alpha f'(r)\neq 0) and $|1-\alpha f(r)| <1$.
But this does not say anything about the convergence of the method.
Then I thought that I can convert this as a functional iteration problem finding fixed point of some function F, this fixed point is the zeros of $f$. That is say $F(x)=x-\alpha f(x)$. Under the condition |F'(r)|=\lambda<1 and then by continuity of F' (assuming f' is cont.) we can get the desired result with the same conditions, 1-\alpha f'(r)\neq 0 and |1-\alpha f'(r)|<1
any help would be appreciated.
-last two lines corrected