Half-Range Fourier Series/Identity Function/Sine

Theorem
Let $\lambda \in \R_{>0}$ be a strictly positive real number.

Let $\map f x: \openint 0 \lambda \to \R$ be the identity function on the open real interval $\openint 0 \lambda$:
 * $\forall x \in \openint 0 \lambda: \map f x = x$

The half-range Fourier sine series for $\map f x$ can be expressed as:

Proof
By definition of half-range Fourier sine series:


 * $(1): \quad \map f x \sim \displaystyle \sum_{n \mathop = 1}^\infty b_n \sin \dfrac {n \pi x} \lambda$

where for all $n \in \Z_{> 0}$:

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


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

as required.