Convergence of Complex Series: Using de Moivre's Theorem | Homework Help

[delefemiaoa]

Homework Statement

Given that Wn = 3^-n cos 2nӨ for n = 1, 2, 3, …, use de Moivre’s theorem to show that

1 + W₁ + W₂ + W₃ + … + W_N-1 = [ 9 – 3 cos2Ө+ 3^-N+1 cos2(N-1)Ө - 3^-N+2 cos2NӨ] / [10 – 6cos2Ө]

Hence show that the infinite series
1 + W₁ + W₂ + W₃ + …
is convergent for all values of Ө, and find the sum to infinity

Please i need help on how to solve this above question. though I have posted it before in my blog but was deleted. I don't know the reason for the deletion. I guessed I should have posted it in homework section

Homework Equations

The Attempt at a Solution

I don't have a clue to this question, I have tried to use A.P and G.P formulas but proved difficult. I need help in order to teach my students preparing for external exams in further maths.

 

[Office_Shredder]

W_n will be the real part of 3^{-n}e^{2n\theta i}. So summing up W_ns is the same thing as taking the real part of a geometric series of terms like that.

 

[delefemiaoa]

I have used ur suggestion,as follows common ration = 3^-1 e^2Өni, first term = 1, however I got stucked.