Power Series of Sine of Odd Theta

Theorem
Let $r \in \R$ such that $\size r < 1$.

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

Then:

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

Hence: