Fourier Series/Identity Function over Minus Pi to Pi

Theorem
For $x \in \openint {-\pi} \pi$:
 * $\displaystyle x = 2 \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n + 1} } n \map \sin {n x}$

Proof
From Odd Power is Odd Function, $x$ is a Odd Function.

By Fourier Series for Odd Function over Symmetric Range, we have:


 * $\displaystyle x \sim \sum_{n \mathop = 1}^\infty b_n \sin n x$

where:

Substituting for $b_n$ in $(1)$:


 * $\displaystyle x = 2 \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n + 1} } n \map \sin {n x}$

as required.