确定常数a,b,c的值,使得当x→0时,ex(1+bx+cx2)=1+ax+o(x3).

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问题 确定常数a,b,c的值,使得当x→0时,ex(1+bx+cx2)=1+ax+o(x3).

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答案由ex=1+x+x2/2+x3/6+o(x3),得ex(1+bx+cx2)=[1+x+x2/2+x3/6+o(x3)](1+bx+cx2)=1+(b+1)x+(b+c+1/2)x2+(b/2+c+1/6)x3+o(x3),所以b+1=a,b+c+1/2=0,b/2+c+1/6=0,即a=1/3,b=-2/3,c=1/6.

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