Question

求这道题的详细解题过程

最好是有详细过程

Answer 1

x->0

e^x = 1+ x+(1/2)x^2+ (1/6)x^3 +(1/24)x^4 +o(x^4)

sinx = x - (1/6)x^3 +(1/120)x^5 +o(x^5)

sinx .e^x

=[x - (1/6)x^3 +(1/120)x^5 +o(x^5)] .[1+ x+(1/2)x^2+ (1/6)x^3 +(1/24)x^4 +o(x^4) ]

=x.[1+ x+(1/2)x^2+ (1/6)x^3 +(1/24)x^4 +o(x^4) ]

-(1/6)x^3.[1+ x+(1/2)x^2+ (1/6)x^3 +(1/24)x^4 +o(x^4) ]

+(1/120)x^5. [1+ x+(1/2)x^2+ (1/6)x^3 +(1/24)x^4 +o(x^4) ] 

=[x+ x^2+(1/2)x^3+ (1/6)x^4 +(1/24)x^5 +o(x^5) ]

+[-(1/6)x^3-(1/6)x^4-(1/12)x^5 +o(x^5) ]

+[(1/120)x^5 +o(x^5) ] 

=x+ x^2+(1/2-1/6)x^3+ (1/6-1/6)x^4+ (1/24-1/12+1/120)x^5 +o(x^5) 

=x+ x^2 +(1/3)x^3 -(1/30)x^5 +o(x^5)

f(x)

= e^x.sinx

=x+ x^2 +(1/3)x^3 -(1/30)x^5 +o(x^5)

追问

e的x次方那项写到o（x⁵）可以吗

追答

可以但没有需要
e^x = 1+ x+(1/2)x^2+ (1/6)x^3 +(1/24)x^4 +（1/120)x^5+o(x^5)
原因
sinx = x - (1/6)x^3 +(1/120)x^5 +o(x^5)
已经保证 e^x 每一项 最小都增加 x^1

追问

谢谢啦

x⁵那项是为啥啊？