Example 8. Convergence: Monotone Decreasing  Find the solution to  .  Use the starting approximation    

Solution 8.

Since   for all   ,  the iteration    will converge to the fixed point  .  In this case,   is said to be an attractive fixed point.  

Since  for all   ,  then the iteration    will not converge to .  In this case,   is said to be a repelling fixed point and the iteration exhibits local divergence.  

Animation 5.  ( Fixed Point Iteration  Fixed Point Iteration ).  Internet hyperlink.

(c) John H. Mathews 2004