请给我∑cosn有界的证明过程~

Question

请给我∑cosn有界的证明过程~

Answer 1

积化和差公式：
2cosn sin(1/2) = sin(n+1/2) - sin(n-1/2)
2cos(n-1) sin(1/2) = sin(n-1/2) - sin(n-3/2)
...
2cos1 sin(1/2) = sin(3/2) - sin(1/2)
以上诸式相加并注意到右边可错项相消,于是
此级数
=( sin(n+1/2)-sin(1/2) ) / ( 2sin(1/2) )
有界.