Starting Values
The error analysis in the previous section shows that if the initial guess is chosen accurately enough, that is, if e[0] is chosen sufficiently small, then the right hand side of enPlus1 will be smaller than e[0] and the sequence {e[n]}, n = 1,... ,Infinity, will decrease to zero. Hence, Newton's method will converge.
Convergence is not guaranteed, however, when the initial guess is not chosen accurately enough.
Up to Newton's Method
Divergence Examples