用级数怎么证欧拉公式?

题目

用级数怎么证欧拉公式?
eix=cos x+isinx

答案

e^ix = 1+ix+(ix)^2/2!+(ix)^3/3!+...
= 1+ix-x^2/2!-ix^3/3!+...
= (1-x^2/2!+...) + i(x-x^3/3!+...)
= cos x + isin x