2(sinx/2)*∑sinkx=(cosx/2)－cos(2n＋1)*x/2这怎么证明
问题描述：
2(sinx/2)*∑sinkx=(cosx/2)－cos(2n＋1)*x/2这怎么证明
答
用积化和差公式
2(sinx/2)*∑sinkx
= ∑2(sinx/2)sinkx
=∑-2( cos(x/2+kx) - cos(x/2-kx) )/2
=∑cos((2k-1)x/2)- cos((2k+1)x/2)
= cos(x/2) - cos(3x/2) + cos (3x/2)-cos(5x/2)....
2(sinx/2)*∑sinkx=(cosx/2)－cos(2n＋1)*x/
注意抵消规律，所以
答
用积化和差公式2sin(a)sin(b)=cos(a-b)-cos(a+b)则2sin(x/2)*sin(x)=cos(x/2)-cos(3x/2)2sin(x/2)*sin(2x)=cos(3x/2)-cos(5x/2)2sin(x/2)*sin(3x)=cos(5x/2)-cos(7x/2)...2sin(x/2)*sin(nx)=cos((n-1/2)x)-cos((2n+1...