自然指数函数的泰勒级数展开：
$$ e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots $$
正弦函数的泰勒级数展开：
$$ \sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots $$
余弦函数的泰勒级数展开:
$$ \cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots $$
自然对数的泰勒级数展开:
$$ \ln(1 + x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots $$
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(define (factorial n)
(if (= n 0) 1
(* n (factorial (- n 1)))))
(define (exp-taylor x terms)
(if (= terms 0)
0
(+ (/ (expt x (- terms 1)) (factorial (- terms 1)))
(exp-taylor x (- terms 1)))))
(define (sin-taylor x terms)
(if (= terms 0)
0
(let ([term-num (sub1 (* 2 terms))])
(+ (* (if (even? terms) -1 1) (/ (expt x term-num) (factorial term-num)))
(sin-taylor x (- terms 1))))))
(define (cos-taylor x terms)
(if (= terms 0)
1
(let ([term-num (sub1 (* 2 terms))])
(+ (* (if (even? terms) 1 -1) (/ (expt x term-num) (factorial term-num)))
(cos-taylor x (- terms 1))))))
(define (ln-taylor x terms)
(if (= terms 0)
0
(+ (* (if (even? terms) -1 1) (/ (expt x terms) terms))
(ln-taylor x (- terms 1)))))
(exp-taylor 1 10)
(sin-taylor 1 10)
(cos-taylor 1 10)
(ln-taylor 0.5 10)
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