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Cos theta cos theta + cos^2 theta cos 2theta + cos^3 theta cos 3theta+.....to infinity.find the sum of the above series

Cos theta cos theta + cos^2 theta cos 2theta + cos^3 theta cos 3theta+.....to infinity.find the sum of the above series
 

Grade:12th pass

1 Answers

Aditya Gupta
2081 Points
5 years ago
let A= Cos theta cos theta + cos^2 theta cos 2theta + cos^3 theta cos 3theta+.....to infinity
B= Cos theta sin theta + cos^2 theta sin 2theta + cos^3 theta sin 3theta+.....to infinity
A+iB= costheta*e^itheta+cos^2theta*e^i2theta+cos^3theta*e^i3theta+.......to infinity
let costheta*e^itheta= y
then A+iB= y+y^2+y^3+..... which is an infinite GP
A+iB= y/(1 – y)
now, A= real part of  y/(1 – y).
now replace y with costheta*e^itheta and expand e^itheta as costheta+isintheta. then you will see that it is a purely imaginary number.
So, A=0

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