Using complex exponentials to prove 1+acostheta

[captainemeric]

Homework Statement

Use complex exponentials to prove 1 + acos(theta) + a^2cos(2theta) + a^3cos(3theta)... = (1 - acos(theta))/(1 - 2acos(theta) + a^2)

Homework Equations

euler's e^itheta/2 +e^-itheta/2=2cos(2theta)

The Attempt at a Solution

a^(n)cos(ntheta) = e^nitheta = e^-nitheta

from there i got the series

(a^n(e^itheta)^n)/2

now from here I think I setup the summation formula but this is where I get stuck. Any help is greatly apprecited.

 

[voko]

Is |a| < 1?

 

[captainemeric]

Yes, a is a real constant and |a| < 1. sorry about that

 

[voko]

$$a^n \cos n\theta = a^n\frac {e^{in\theta} + e^{-in\theta}} {2}<br /> = \frac {a^ne^{in\theta} + a^ne^{-in\theta}} {2}<br /> = \frac {p^n + q^n} {2}<br /> \\ p = (\ln a)e^{i\theta}, \ |p| < 1<br /> \\ q = (\ln a)e^{-i\theta}, \ |q| < 1$$

What is the sum of $p^n$ and $q^n$?

 

[captainemeric]

That makes sense. That will then give me a real and an imaginary result of which I take the real I believe. Also, I apologize for the typo on the first post.

 

[voko]

Well, you can take the real part, but the sum is real anyway.