sicp-all-tasks/sicp/3_002e59

Exercise 3.59: In 2.5.3 we saw how
to implement a polynomial arithmetic system representing polynomials as lists
of terms.  In a similar way, we can work with 
power series, such as

  
    
      
        
          e
          x
        
      
      
        =
      
      
        1
        +
        x
        +
        
          1
          2
        
        
          x
          2
        
        +
        
          1
          
            3
            ⋅
            2
          
        
        
          x
          3
        
        +
        
          1
          
            4
            ⋅
            3
            ⋅
            2
          
        
        
          x
          4
        
        +
        …
        ,
      
    
    
      
        cos
        ⁡
        x
      
      
        =
      
      
        1
        −
        
          1
          2
        
        
          x
          2
        
        +
        
          1
          
            4
            ⋅
            3
            ⋅
            2
          
        
        
          x
          4
        
        −
        …
        ,
      
    
    
      
        sin
        ⁡
        x
      
      
        =
      
      
        x
        −
        
          1
          
            3
            ⋅
            2
          
        
        
          x
          3
        
        +
        
          1
          
            5
            ⋅
            4
            ⋅
            3
            ⋅
            2
          
        
        
          x
          5
        
        −
        …
      
    
  

represented as infinite streams.  We will represent the series 
  
    
      a
      0
    
    +
    
      a
      1
    
    x
  
  +
  
    
      a
      2
    
    
      x
      2
    
  
  +
  
    
      a
      3
    
    
      x
      3
    
    +
    …
  
 as the stream whose
elements are the coefficients 
  
    a
    0
  
, 
  
    a
    1
  
, 
  
    a
    2
  
, 
  
    a
    3
  
, ….


 The integral of the series 
  
    
      a
      0
    
    +
    
      a
      1
    
    x
  
  +
  
    
      a
      2
    
    
      x
      2
    
  
  +
  
    
      a
      3
    
    
      x
      3
    
    +
    …
  
 is the series

  c
  +
  
    
      a
      0
    
    x
  
  +
  
    
      1
      2
    
    
      a
      1
    
    
      x
      2
    
  
  +
  
    
      1
      3
    
    
      a
      2
    
    
      x
      3
    
  
  +
  
    
      1
      4
    
    
      a
      3
    
    
      x
      4
    
    +
    …
    ,
  

where 
  c
 is any constant.  Define a procedure integrate-series that
takes as input a stream 
  
    a
    0
  
, 
  
    a
    1
  
, 
  
    a
    2
  
, … representing a power
series and returns the stream 
  
    a
    0
  
, 
  
    
      
        1
        2
      
    
    
      a
      1
    
  
, 
  
    
      
        1
        3
      
    
    
      a
      2
    
  
, … of
coefficients of the non-constant terms of the integral of the series.  (Since
the result has no constant term, it doesn’t represent a power series; when we
use integrate-series, we will cons on the appropriate constant.)

 The function 
  
    x
    ↦
    
      e
      x
    
  
 is its own derivative.  This implies that

  
    e
    x
  
 and the integral of 
  
    e
    x
  
 are the same series, except for the
constant term, which is 
  
    
      e
      0
    
    =
    1
  
.  Accordingly, we can generate the series
for 
  
    e
    x
  
 as


(define exp-series
  (cons-stream 
   1 (integrate-series exp-series)))

Show how to generate the series for sine and cosine, starting from the facts
that the derivative of sine is cosine and the derivative of cosine is the
negative of sine:


(define cosine-series 
  (cons-stream 1 ⟨??⟩))

(define sine-series
  (cons-stream 0 ⟨??⟩))