YLL討論網

從昌爸轉貼的,因為我也不會,所以貼到yll來求救.

求值式
1+ 2cosA +3cos2A +4cos3A +......+48cos47A +49cos48A =?

某位叫mike的回答,
[缺一條件------>A=(2/49)pi
解答
負的二分之四十九------>(-49/2)
令w=cosA+isinA
Sum=1+2w+3w^2+4w^3+......+48w^47+49w^48
-)w*Sum=w+2w^2+3w^3+4w^4+......+48w^48+49w^49
-----------------------------------------------------
(1-w)*S=1+w+w^2+3w^3+4w^4+......+48w^48-49w^49

S=(-49w^49)/(1-w)......不會解啦!........(煙霧彈......)]

(1-w)*S=1+w+w^2+w^3+.......+w^48-49w^49
(1-w)*S=[(1-w^49)/(1-w)]-49w^49..............(1)
      w^49=1代入(1)得(1-w)*S=-49
   S=-49/(1-w)
     =-49/(1-cos2p/49-isin2p/49)
     =-49/{1-[1-2(sinp/49)^2]-2isinp/49*cosp/49}
     =[-49/(2sinp/49)]*[1/(sinp/49-icosp/49)]
     =[-49/(2sinp/49)]*[sinp/49+icosp/49]
     =-49/2-(49/2)i*cotp/49
      
所以1+2cosA+3cos2A+........+49cos48A=-49/2
     2sinA+3sin2A+..........+49sin48A=(-49/2)*cotp/49