I need your help, i am busy with the Newton's method, finding roots. At the moment there is this polynomial $P(x)=x^2+1$.
draw the graph 
So normally we need to find the spot $(x,0)$ so then we can choose a point $x_0$ and start finding the first iterates using $$x_{n+1} = x_n − \frac{P(x)}{P'(x)},$$ but in this case there in no $(x,0)$ because as we can see the graph does not touch the axis $x$.
the question is to compute the first iterates with any initial condition $x_0$. Is it convergent? Why (justify the answer)?
What i did is to choose $x_0=1$ and started doing the iterates, but i did not end up somewhere which it will be helpful. Somebody any idea?
thanks
HINT Your iteration is $$ x_{n+1} = x_n - \frac{x_n^2+1}{2x_n} = \frac{x_n^2-1}{2x_n} = \frac{x_n+(-1/x_n)}{2}. $$ In other words, each consecutive iterate is the average of the current position and its negative multiplicative inverse. What happens under such iterations, e.g. if $x_n$ is large or around 0 or 1? Try out a couple different starting points...
UPDATE
For $x_0=1$ you have $x_1=0$ and the iteration stops. For any other ones, you iterate decreasing magnitude until you hit a point inside $(-1,1)$ and then the next iterate jumps high, and the closer you are to $0$, the more you jump. This is a non-convergent system.
In numerical analysis, when you practically rootfind, you ride Newton while it decreases absolute value of the estimate and then switch to something like the bisection method.