Determine the Fourier expansion of the function $f(x)=\sin \lambda x$, where $-\pi \leqslant x \leqslant \pi$, in the two cases where $\lambda$ is an integer and $\lambda$ is a real non-integer.

Using the Parseval identity in the case $\lambda=\frac{1}{2}$, find an explicit expression for the sum

$$\sum_{n=1}^{\infty} \frac{n^{2}}{\left(4 n^{2}-1\right)^{2}}$$