Does the following series converge?
$f(x)=\displaystyle\sum_{n=1}^{\infty}\frac{(n\pi)^2}{1+n\pi} \big(a_n\sin(n\pi x) - b_n\cos(n\pi x) \big)^2$
where $a_n= \displaystyle\frac{1}{\pi}\displaystyle\int_0^{2\pi}f(x)\sin(n x)dx$ and $b_n= \displaystyle\frac{1}{\pi}\displaystyle\int_0^{2\pi}f(x)\cos(n x)dx$.
For $\;x=0\;$ we get
$$\sum_{n=1}^\infty\frac{n^2\pi^2}{1+n\pi}b_n^2$$
so unless $\;b_n\;$ is, for example, a sequence converging to zero pretty quick (at least as fast as $\;\frac1{n^{1+\epsilon}}\;$ , for some $\;\epsilon>0\,$) , the sum diverges.
Meaning: without knowing what the $\;a_n,\,b_n\;$ are, one can't say much about this series.