Exercise 4.4
A stream that converges to the square root a number **

It is well-known that the square root function can be implemented by use of Newton's method. In case you need some background, you can consult this video.

If we need to find sqrt(x) based on an initial quess g, we can iteratively improve our guess by this simple function:

(define (improve-sqrt-guess guess x)
  (/ (+ guess (/ x guess)) 2))

Produce a stream of real numbers, starting with the initial guess 1.0, that will converge towards sqrt(x).

You may find the following stream variant of map useful:

  (define (map-stream f stream)
    (cond ((empty-stream? stream) the-empty-stream)
          (else (cons-stream (f (head stream)) (map-stream f (tail stream))))))

Here is all you need to get started:

(define (improve-sqrt-guess guess x)
 (/ (+ guess (/ x guess)) 2))

(define-syntax cons-stream
  (syntax-rules ()
    ((cons-stream x y)
     (cons x (delay y)))))

(define head car)

(define (tail stream) (force (cdr stream)))

(define empty-stream? null?)

(define the-empty-stream '())

(define (stream-section n stream)
  (cond ((= n 0) '())
        (else (cons (head stream)
              (stream-section 
                (- n 1)
               (tail stream))))))


(define (map-stream f stream)
  (cond ((empty-stream? stream) the-empty-stream)
        (else (cons-stream (f (head stream)) (map-stream f (tail stream))))))