Computing a Geometric Power Series with Cosine Terms

[quasar_4]

Homework Statement

Compute $$\sum_{n=0}^{\infty} p^n cos(3nx)$$ for $$\abs{p} \textless 1$$, where $$p \in \mathbb{R}$$.

Homework Equations

The Attempt at a Solution

I was thinking that maybe this could be approached as a telescoping series, but I'm not really sure if it is. Would that be the most expedient approach? Clearly it isn't geometric, and I'm not sure how to find the sum of a general power series.

 

[quasar_4]

No one? Someone has to know. There are a lot of clever people on this forum.

 

[quasar_4]

What if I rearrange it using the fact that cos(3nx) = 1/2(exp(3nix)-exp(-3inx)). Could I then write $$\sum_{n=0}^{\infty} (pe^{3xi})^n + \sum_{n=0}^\infty (pe^{-3ix})^n$$ and try to work from there? (ie, is that valid?)

 

[quasar_4]

Ok, I was being silly, as usual - it is actually geometric after all.

Thank you to me for figuring out this problem.