Sum of Infinite Series of Product of Power and Sine

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

Let:
 * $S \left({x}\right) := \displaystyle \sum_{n \mathop = 1}^\infty a^n \sin n x$

Then:
 * $S \left({x}\right) = \dfrac {a \sin x} {1 - 2 a \cos x + a^2}$

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

Hence:

Also see

 * Sum of Series of Product of Power and Cosine