求证e^i(4π/n)+e^i(8π/n)+...+e^i4(n-1)π/n+e^i(4nπ/n)=0e^i(4π/n)+e^i(8π/n)+...+e^i(4(n-1)π/n)+e^i(4nπ/n)=e^i(4π/n)[1-e^i（4nπ/n）]/[1-e^4π]=0如何证明

Question

求证e^【i(4π/n) 】+ e^【i(8π/n) 】+ ...+ e^i【4(n-1)π/n】 + e^【i(4nπ/n)】 =0
e^i(4π/n) + e^i(8π/n) + ...+ e^i(4(n-1)π/n) + e^i(4nπ/n)
= e^i(4π/n)[1-e^i（4nπ/n）]/[1-e^4π]
= 0
如何证明

Answer 1

你只差一步了,注意中括号里面的第二项=1：
e^i(4nπ/n) =e^(4πi)
=cos4π+isin4π
=1
所以中括号=0,从而结论成立.