确定a,b的值,使得当x→0时,f(x)=x-(a+bcosx)sinx成为x^5的同价无穷小量

Question

确定a,b的值,使得当x→0时,f(x)=x-(a+bcosx)sinx成为x^5的同价无穷小量

Answer 1

分别将cosx与sinx在0点泰勒展开.
因为只要考虑x^5的同阶无穷小量,根据原式,我们只需作如下近似展开：
cosx=1-x^2/2+x^4/24+o(x^4)
sinx=x-x^3/6+x^5/120+o(x^5)
代入原式,我们有
f(x)=x-[a+b(1-x^2/2+x^4/24+o(x^4))]*[x-x^3/6+x^5/120+o(x^5)]
=(1-a-b)*x - (a/6 + 2b/3)*x^3 + (a/120 + 2b/15)*x^5 + o(x^5)
所以,1-a-b=0,a/6 + 2b/3=0,a/120 + 2b/15≠0.
解得,a=4/3,b=-1/3.