1 ;; The first three lines of this file were inserted by DrScheme. They record metadata
2 ;; about the language level of this file in a form that our tools can easily process.
3 #reader(lib "htdp-intermediate-reader.ss" "lang")((modname |23.3|) (read-case-sensitive #t) (teachpacks ((lib "draw.ss" "teachpack" "htdp") (lib "arrow.ss" "teachpack" "htdp") (lib "gui.ss" "teachpack" "htdp"))) (htdp-settings #(#t constructor repeating-decimal #f #t none #f ((lib "draw.ss" "teachpack" "htdp") (lib "arrow.ss" "teachpack" "htdp") (lib "gui.ss" "teachpack" "htdp")))))
4 ;! : N -> number
5 ;Computes the factorial of n.
6 
7 (define (! n)
8   (cond
9     [(zero? n) 1]
10     [else (* n (! (sub1 n)))]))
11 
12 ;series : (N[>=0] -> number) N[>=0] -> number
13 ;Given a-term and n, determine the sum of the first n-th terms in the sequence a-term.
14 
15 (define (series a-term n)
16   (cond
17     [(zero? n) (a-term n)]
18     [else (+ (a-term n)
19              (series a-term (sub1 n)))]))
20 
21 ;;e-taylor : N -> number
22 ;;Given i, find the i-th term of the Taylor series e^x = 1 + x + (x^2)/2! ...
23 
24 ;e-power : number N -> number
25 ;e-power computes the x-th power of e by summing up n terms of the Taylor sequence.
26 
27 (define (e-power x n)
28   (local ((define (e-taylor i)
29             (/ (expt x i) (! i))))
30     (series e-taylor n)))
31 
32 #|
33 
34 (define (inexact-e-power x n)
35   (exact->inexact (e-power x n)))
36 
37 (inexact-e-power 1 1000)
38 (exp 1)
39 (- (inexact-e-power 1 100) (exp 1))
40 |#
41 
42 ;ln-taylor : N -> number
43 ;Computes the n-th term of the taylor sequence for ln (x).
44 
45 ;ln-x : number N -> number
46 ;Computes ln(x) by summing the first n-th terms of the Taylor sequence.
47 
48 (define (ln-x x n)
49   (local ((define (ln-taylor i)
50             (* 2 (/ (expt (/ (- x 1)
51                              (+ x 1))
52                           (+ (* 2 i) 1))
53                     (+ (* 2 i) 1)))))
54     (series ln-taylor n)))
55 
56 ;sin-x : number N -> number
57 ;Determines sin(x) by summing the first n-th terms of the Taylor sequence.
58 
59 ;sin-taylor : N -> number
60 ;Returns the i-th term of the Taylor sequence for sin(x).
61 
62 (define (sin-x x n)
63   (local ((define (sin-taylor i)
64             (* (/ (expt x (+ (* 2 i) 1))
65                   (! (+ (* 2 i) 1)))
66                (cond
67                  [(even? i) 1]
68                  [(odd? i) -1]))))
69     (series sin-taylor n)))
70 
71 ;pi-greg : N -> number
72 ;Calculates pi by summing the first n-th terms of the Taylor sequence.
73 
74 ;greg : N -> number
75 ;Given i, find the n-th term of the greg series.
76 
77 (define (pi-greg n)
78   (local ((define (greg i)
79             (* 4 (* (/ 1
80                        (+ (* 2 i)
81                           1))
82                     (cond
83                       [(even? i) 1]
84                       [(odd? i) -1])))))
85     (series greg n)))