#lang sicp

; First, repeat the algorithm in the book
(define (sum term a next b)
  (if (> a b)
      0
      (+ (term a)
         (sum term (next a) next b))))

(define (pi-sum a b)
  (define (pi-term x)
    (/ 1.0 (* x (+ x 2))))
  (define (pi-next x)
    (+ x 4))
  (sum pi-term a pi-next b))

(* 8 (pi-sum 1 1000))

; reimplementing it iteratively
(define (sum-iter term a next b)
  (define (iter a result)
    (if (> a b)
        result
        (iter (next a) (+ result (term a)))))
  (iter a 0))

(define (pi-sum-iter a b)
  (define (pi-term x)
    (/ 1.0 (* x (+ x 2))))
  (define (pi-next x)
    (+ x 4))
  (sum-iter pi-term a pi-next b))

(* 8 (pi-sum-iter 1 1000))