De Moivre's Theorem and Power Series

1. Jan 25, 2015

machofan

1. The problem statement, all variables and given/known data
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))

2. Relevant equations

3. 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.

2. Jan 25, 2015

MostlyHarmless

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