sicp-ex-3.65

  
  
 (define (ln2-summands n) 
   (cons-stream (/ 1.0 n) 
                (stream-map - (ln2-summands (+ n 1))))) 
 (define ln2-stream 
   (partial-sums (ln2-summands 1))) 

Just wondering why not using recurrence relation to define summands, for instance for ln:

  
 (define ln-summands 
   (cons-stream 1 
    (stream-map (lambda (x) 
                  (* (if (> x 0) -1 1) 
                     (/ 1 (+ (denominator x) 1)))) 
                ln-summands))) 
  
 (define ln-stream 
  (stream-map exact->inexact (partial-sums ln-summands)))  
  

  
  
 (define (sub-streams stream1 stream2) 
     (stream-map - stream1 stream2)) 
  
 (define (one-order-difference stream) 
     (sub-streams (stream-cdr stream) stream)) 
  
 (define (euler-transform s) 
     (sub-streams 
         (stream-cdr (stream-cdr s)) 
         (stream-map 
             / 
             (stream-map square (one-order-difference (stream-cdr s))) 
             (one-order-difference (one-order-difference s))))) 
  
 (define (ln2-summands n) 
     (cons-stream 
         (/ 1.0 n) 
         (stream-map - (ln2-summands (+ n 1))))) 
  
 (define ln2-stream 
     (partial-sums (ln2-summands 1))) 
  
 (define (show-streams n . streams) 
     (if (accumulate 
             (lambda (a b) (or a b)) 
             #f 
             (cons (= n 0) (map stream-null? streams))) 
         (newline) 
         (begin 
             (display-line (map stream-car streams)) 
             (apply show-streams (cons (- n 1) (map stream-cdr streams)))))) 
  
 (define (make-ln2-one-order-difference-tableau) 
     (let ((ln2-tableau (make-tableau euler-transform ln2-stream))) 
         (lambda (i) (one-order-difference (stream-ref ln2-tableau i))))) 
  
 (define ln2-oodt (make-ln2-one-order-difference-tableau)) 
  
 (show-streams 12 (ln2-oodt 4) (ln2-oodt 5) (ln2-oodt 6)) 
 ; It can be seen that the convergence rate of this series increases by 2-3 orders of magnitude every time it passes through Euler transformation