De Moivre's Theorem and Power Series

[machofan]

Homework Statement

Hi I'm stuck with the following question:

Use de Moivre's Theorem and your knowledge of power series to show:

1/1(1/2^1)cos(θ)+1/2(1/2^2)cos(2θ)+1/3(1/2^3)cos(3θ)+ ... = log(2)-1/2*log(5-4cos(θ))

Homework Equations

The Attempt at a Solution

I have already established the series to be (1/2)(∑((eiθ/2)^n/n) + ∑((e-iθ/2)n)/n) and evaluated the two series as a function of a natural logarithm ∑(x^n/n). But I'm not sure where to go from here, any help is much appreciated thanks.

 

[MostlyHarmless]

If you're stuck, try working it from the other direction. Start with $log(2)- (1/2)log(5-4cos\theta)$.