Sum of Infinite Series of Product of Power and Sine

Theorem
Let $r \in \R$ such that $\left\lvert{r}\right\rvert < 1$.

Let $x \in \R$ such that $x \ne 2 m \pi$ for any $m \in \Z$.

Then:

Proof
From Euler's Formula:
 * $e^{i \theta} = \cos \theta + i \sin \theta$

Hence:

It is noted that when $x$ is a multiple of $2 \pi$ then:
 * $1 - 2 r \cos x + r^2 = 1 - 2 + 1 = 0$

leaving the undefined.

Also see

 * Sum of Infinite Series of Product of Power and Cosine


 * Sum of Series of Product of Power and Sine
 * Sum of Series of Product of Power and Cosine