求(sinx)^3带皮亚诺型余项的马克劳林公式

Question

求(sin x)^3带皮亚诺型余项的马克劳林公式

Answer 1

先化成一次的式子
(sinx)^3
=sinx(1-cos2x)/2
=(1/2)sinx - (1/2)sinxcos2x
积化和差
=(1/2)sinx - (1/4)(sin3x-sinx)
=(3/4)sinx - (1/4)sin3x
展开
=(3/4)(x-x^3/3! + x^5/5!+...+(-1)^n*x^(2n+1)/(2n+1)! + o(x^(2n+1))
- (1/4)(3x - (3x)^3/3! +...+(-1)^n*(3x)^(2n+1)/(2n+1)! + o(x^(2n+1)))
=x^3 -(1/2)x^5+...+ (3/4)*(-1)^n * (1-3^(2n)) / (2n+1)!+o(x^(2n+1))