sol-1.31.tex

\begitems\style a
* The `product` procedure can be defined as follows:
\begtt\scm
(define (product term a next b)
  (define (iter a result)
    (if (> a b)
        result
        (iter (next a) (* result (term a)))))
  (iter a 1))
\endtt
Then we can define `factorial` and `pi-product`
\begtt\scm
(define (factorial n)
  (define (identity x) x)
  (define (inc x) (+ 1 x))
  (product identity 1 inc n))

(define (pi-product n)
  (define (pi-term x)
    (/ (* x (+ 2 x))
       (square (+ 1 x))))
  (define (pi-next x)
    (+ 2 x))
  (product pi-term 2.0 pi-next (* 2 n)))
\endtt
and compute some approximations of $\pi$:
\begtt\scm
(* 4 (pi-product 100))
;Value: 3.149378473168601
(* 4 (pi-product 1000))
;Value: 3.1423773650938855
(* 4 (pi-product 10000))
;Value: 3.1416711865345
\endtt
* The above defined `product` procedure generates an iterative process;  here is one that generates a recursive process:
\begtt\scm
(define (product term a next b)
  (if (> a b)
      1
      (* (term a)
         (product term (next a) next b))))
\endtt
\enditems