Test of quadratic convergence.  How does one tell when it is occuring?

Newtons method on

f[x_] = (x^2) - 14;

ListOfIterates = 
      NestList[N[(# - f[#]/(f')[#]) &, 30], 1, 
        8]
RotateLeft[(ListOfIterates - 2)]/(((ListOfIterates - 2))^2)

ListOfIterates - 2

RotateLeft[ListOfIterates] - ListOfIterates

ListPlot[Out[21], PlotJoined -> True];

Results:When newtons method is working properly the approximations head steadly to the root.  After several iterations ( say 5 the number is fairly small ) the quotent above can be used to test for quadratic convergence.  It should limit to an number.  Another sanity check for a newton method is that the iterates are getting closer and closer to each other.  If not the method may not be converging.