sicp-all-tasks/sicp/1_002e45

Exercise 1.45: We saw in 1.3.3
that attempting to compute square roots by naively finding a fixed point of

  
    y
    ↦
    x
    
      /
    
    y
  
 does not converge, and that this can be fixed by average
damping.  The same method works for finding cube roots as fixed points of the
average-damped 
  
    y
    ↦
    x
    
      /
    
    
      y
      2
    
  
.  Unfortunately, the process does not
work for fourth roots—a single average damp is not enough to make a
fixed-point search for 
  
    y
    ↦
    x
    
      /
    
    
      y
      3
    
  
 converge.  On the other hand, if
we average damp twice (i.e., use the average damp of the average damp of 

  
    y
    ↦
    x
    
      /
    
    
      y
      3
    
  
) the fixed-point search does converge.  Do some experiments
to determine how many average damps are required to compute 
  
    n
    
      th
    
  
 roots as a
fixed-point search based upon repeated average damping of 
  
    y
    ↦
    x
    
      /
    
    
      y
      
        
        n
        −
        1
      
    
  
.  
Use this to implement a simple procedure for computing

  
    n
    
      th
    
  
 roots using fixed-point, average-damp, and the
repeated procedure of Exercise 1.43.  Assume that any arithmetic
operations you need are available as primitives.