Thread: Newton method help

I have an exam on Tuesday 27th. I need to know well the bisection method, secant method, newton method and a lot more.
I have a sheet of around 25 exercises. I could do some pretty easy, but others are too hard for me.
I have a lot of questions in relation with these methods. (That I will post in other threads if I feel the need to).
Here comes an exercise I couldn't do.
1)Demonstrate that if f is a function that has a double root in a, then the Newton method applied to f has an order of convergence of 1.
Demonstrate also that the modified Newton has a convergence order of 2.
2)Study what happens if .
As p is not defined, I guess it's the number of multiple roots.
End of the problem.
My approach was to write as . and . Therefore the Newton method will converges regardless of the starting point.
From it, I supposed . I calculated , which is worth . Also and lastly .
So our intuition tells us that the method converges to a. But how can I determine the order? And how can I prove that it effectively converges to a?
I'd be glad if you could start me up for the common Newton method, this way I hope to finish the rest alone.

I calculated and guessed well . In fact I noticed the coefficient in front of can be written as a sequence that tends to 1 when n tends to positive infinite.
I found that . I set . I'm completely despaired. I need help on this problem. How can I prove that the sequence tends to ? The first terms are , , , , etc. I know it's a famous sequence, where each step is half closer to 1 than the precedent.
But I need to be rigorous, I can't say " I'm seeing that the coefficients of the method are like this sequence". I must prove it and I'm not able.
Is there any other way I'm not seeing to do the first question? I'm lost.