Higher-order procedures

a. The sum procedure is only the simplest of a vast number of similar abstractions that can be captured as higher-order procedures. Write an analogous procedure called product that returns the product of the values of a function at points over a given range. Show how to define factorial in terms of product. Also use product to compute approximations to using the formula

    π     2·4·4·6·6·8···
    -  =  --------------
    4     3·3·5·5·7·7···

b. If your product procedure generates a recursive process, write one that generates an iterative process. If it generates an iterative process, write one that generates a recursive process.


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(define (inc n) (+ n 1))

(define (square x) (* x x))

(define (identity x) x)

(check-equal? (product square 1 inc 3) 36)
(check-equal? (product identity 3 inc 5) 60)
(check-equal? (factorial 5) 120)