<s>[INST] Show equality of a given function with a series in $ℝ$ Show that: $$2x\cos x-\sin x=4\sum_{n=2}^\infty \frac{(-1)^n}{n^2-1}\sin(nx)$$
Supposedly, this can be proved by using Fourier series, by choosing the right function but I have been thus far unable to do so. Any hints will be welcome, thanks in advance!
 [/INST] By using the Fourier sine series 
\begin{align}
f(x) &= \sum_{n=1}^{\infty} B_{n} \, \sin(nx)  \hspace{15mm} x \in [0,\pi] \\
B_{n} &= \frac{2}{\pi} \, \int_{0}^{\pi} f(x) \, \sin(nx) \, dx
\end{align}
the function $f(x) = a x \cos(x) + b \sin(x)$ can be seen to be represented by
\begin{align}
a x \cos(x) + b \sin(x) = \frac{2b - a}{2} + 2a \, \sum_{n=2}^{\infty} \frac{(-1)^{n} \, n}{n^{2} -1} \, \sin(nx).
\end{align}
This is obtained by making use of 
\begin{align}
B_{1} &= \frac{2}{\pi} \, \int_{0}^{\pi} (ax \cos(x) + b \sin(x)) \, \sin(x) \, dx = \frac{2b-a}{2}
\end{align}
and calculating the general case for $n \geq 2$. Now choosing $a$ and $b$ as 
$a=2$ and $b=1$ then the reduction provides
\begin{align}
2 x \cos(x) + \sin(x) = 4 \, \sum_{n=1}^{\infty} \frac{(-1)^{n} \, n}{n^{2}-1} \, \sin(n x).
\end{align}
</s>