Ex: 3.53 - 3.62

;some code from book
(define (stream-enumerate-interval low high)
 (if (> low high)
     the-empty-stream
     (cons-stream
      low
      (stream-enumerate-interval (+ low 1) high))))
(define (display-stream s)
 (stream-for-each display-line s))
(define (display-line x)
 (newline)
 (display x))
(define (add-streams s1 s2)
 (stream-map + s1 s2))
(define (integers-starting-from n)
 (cons-stream n (integers-starting-from (+ n 1))))
(define integers (integers-starting-from 1))
(define (scale-stream stream factor)
 (stream-map (lambda (x) (* x factor)) stream))

Ex 3.53
Given definition defines a stream whose first term is 1, and next terms are calculated by adding previous term to itself. So the resulting stream is..
1,2,4,8,16...
Lets Verify...

(define s (cons-stream 1 (add-streams s s)))
1 ]=> (stream-ref s 0)
;Value: 1
1 ]=> (stream-ref s 1)
;Value: 2
1 ]=> (stream-ref s 2)
;Value: 4
1 ]=> (stream-ref s 3)
;Value: 8
1 ]=> (stream-ref s 4)
;Value: 16
1 ]=> (stream-ref s 5)
;Value: 32

Ex - 3.54

(define (mul-streams s1 s2)
 (stream-map * s1 s2))
(define factorials (cons-stream 1 (mul-streams integers factorials)))

Ex-3.55
Partial-sums of a stream is a stream whose first term is that of the input stream and next terms are calculated by adding next term of the input stream and previous term of the Partial-sums of the stream.

(define (partial-sums s)
 (cons-stream (stream-car s)
              (add-streams (stream-cdr s) (partial-sums s))))

;Verify
(define s (partial-sums integers))
1 ]=> (stream-ref s 0)
;Value: 1
1 ]=> (stream-ref s 1)
;Value: 3
1 ]=> (stream-ref s 2)
;Value: 6
1 ]=> (stream-ref s 3)
;Value: 10
1 ]=> (stream-ref s 4)
;Value: 15
1 ]=> (stream-ref s 5)
;Value: 21

Ex-3.56

(define (merge s1 s2)
 (cond ((stream-null? s1) s2)
       ((stream-null? s2) s1)
       (else
        (let ((s1car (stream-car s1))
   (s2car (stream-car s2)))
(cond ((<> s1car s2car)
      (cons-stream s2car (merge s1 (stream-cdr s2))))
     (else
      (cons-stream s1car
                   (merge (stream-cdr s1)
                          (stream-cdr s2)))))))))
(define S (cons-stream 1 (merge
                         (merge (scale-stream s 2) (scale-stream s 3))
                         (scale-stream s 5))))

Ex-3.57



Ex-3.58
first element is (quotient (* num radix) den)
and next elements are calculated recursively from
num = (remainder (* num radix) den) with same radix and den.

(define (expand num den radix)
 (cons-stream
  (quotient (* num radix) den)
  (expand (remainder (* num radix) den) den radix)))

(expand 1 7 10) should be
1 4 2 8 5 7....

;From the interpreter
1 ]=> (define s (expand 1 7 10))
;Value: s
1 ]=> (stream-ref s 0)
;Value: 1
1 ]=> (stream-ref s 1)
;Value: 4
1 ]=> (stream-ref s 2)
;Value: 2
1 ]=> (stream-ref s 3)
;Value: 8
1 ]=> (stream-ref s 4)
;Value: 5

Ex-3.59

;(a)
;inverted-integers = 1 1/2 1/3 1/4........
(define inverted-integers (stream-map (lambda (x) (/ 1 x)) integers))
(define (integrate-series s)
 (mul-streams s inverted-integers))

;(b)
(define cosine-series
 (cons-stream 1 (integrate-series (stream-map (lambda (x) (- x)) sine-series))))
(define sine-series
 (cons-stream 0 (integrate-series cosine-series)))

Ex-3.60

(define (mul-series s1 s2)
 (cons-stream (* (stream-car s1) (stream-car s2))
              (add-streams
               (scale-stream (stream-cdr s1) (stream-car s2))
               (mul-series s1 (stream-cdr s2)))))

(define one (add-streams
            (mul-series sine-series sine-series)
            (mul-series cosine-series cosine-series)))

;verification, only constant term is 1
;rest all is 0
1 ]=> (stream-ref one 0)
;Value: 1
1 ]=> (stream-ref one 1)
;Value: 0
1 ]=> (stream-ref one 2)
;Value: 0
1 ]=> (stream-ref one 3)
;Value: 0
1 ]=> (stream-ref one 4)
;Value: 0
1 ]=> (stream-ref one 5)
;Value: 0

Ex-3.61

(define (invert-unit-series s)
 (cons-stream 1
              (stream-map (lambda (x) (- x))
                          (mul-series
                           (stream-cdr s)
                           (invert-unit-series s)))))

;Test
;secX = 1/cosX
(define sec-series (invert-unit-series cosine-series))
1 ]=> (stream-ref cosec-series 0)
;Value: 1
1 ]=> (stream-ref cosec-series 1)
;Value: 0
1 ]=> (stream-ref cosec-series 2)
;Value: 1/2
1 ]=> (stream-ref cosec-series 3)
;Value: 0
1 ]=> (stream-ref cosec-series 4)
;Value: 5/24
1 ]=> (stream-ref cosec-series 6)
;Value: 61/720
1 ]=> (stream-ref cosec-series 8)
;Value: 277/8064

Ex-3.62

;one-zeros = 1 0 0 0 0 0........
(define zeros (cons-stream 0 zeros))
(define one-zeros
 (cons-stream 1 zeros))

;s1/s2 = s1.(1/s2)
;we'll have to bring out the constant term from s2 so as
;to make the constant term 1
(define (div-series s1 s2)
              (scale-stream
               (mul-series s1
                           (invert-unit-series
                            (scale-stream s2 (/ 1 (stream-car s2)))))
               (/ (stream-car s2))))
              
;tanX = sinX/cosX
(define tangent-series
 (div-series sine-series cosine-series))

1 ]=> (stream-ref tangent-series 0)
;Value: 0
1 ]=> (stream-ref tangent-series 1)
;Value: 1
1 ]=> (stream-ref tangent-series 2)
;Value: 0
1 ]=> (stream-ref tangent-series 3)
;Value: 1/3
1 ]=> (stream-ref tangent-series 4)
;Value: 0
1 ]=> (stream-ref tangent-series 5)
;Value: 2/15
1 ]=> (stream-ref tangent-series 7)
;Value: 17/315
1 ]=> (stream-ref tangent-series 9)
;Value: 62/2835
1 ]=> (stream-ref tangent-series 11)
;Value: 1382/155925