Prove quadratic convergence and limit

Interesting problem. I don't know a smart way to do it, so I just dive right in and calculate the series up to two terms around x_i (I won't bother to indicate the higher order terms):

[tex]
g(z) = \frac{1}{2} g''\left(x_i\right) \left(z-x_i\right){}^2+g'\left(x_i\right) \left(z-x_i\right)+g\left(x_i\right)
[/tex]

[tex]
g(g(z)) = \frac{1}{2} \left(g''\left(g\left(x_i\right)\right) g'\left(x_i\right){}^2+g'\left(g\left(x_i\right)\right) g''\left(x_i\right)\right)
\left(z-x_i\right){}^2+g'\left(g\left(x_i\right)\right) g'\left(x_i\right) \left(z-x_i\right)+g\left(g\left(x_i\right)\right)

[/tex]

Now if we go ahead and replace these into your expression and use the fact that g(g(x_i) = g(x_i) = x_i we get the following for z_{n + 1}:

[tex]\frac{g''\left(x_i\right){}^2 \left(x_i-z\right){}^3+4 x_i \left(g'\left(x_i\right)-1\right){}^2+2 \left(z-x_i\right) \left(g'\left(x_i\right)-1\right) \left(2 x_i+z
g'\left(x_i\right)\right) g''\left(x_i\right)}{4 \left(g'\left(x_i\right)-1\right){}^2+2 \left(z-x_i\right)
\left(g'\left(x_i\right){}^2+g'\left(x_i\right)-2\right) g''\left(x_i\right)}
[/tex]

Note: I dropped the subscript z_n because of formatting problems with LaTeX in these forums.

I do not know how to simplify the expressions involving the derivatives, but maybe this is already to the point where simplifying into a particular form might make it easy to see the convergence.