I am given the Sin series expansion: (1)$$x=2\sum_{1}^{\infty}\frac{(-1)^{n+1}}{n}Sin(nx)$$
But how do I deal with a solution given by: (2)$$S(x)=\sum_{2}^{\infty}(-1)^{n}\frac{n^{3}}{n^{4}+1}Sin(nx)$$
However, as can be seen, the above series converges poorly; like 1/n
I need to get the above Sin series expansion(2) into the form $$S(x)=\frac{-x}{2}|periodic extension +\frac{Sin(x)}{2}-\sum_{1}^{\infty}\frac{(-1)^{n}}{n(n^{4}+1)}Sin(nx)$$
Been looking at this for a couple of hours but couldn't figure out how to.
Thanks in advance