Note that Let $a_n=\int_{-\pi}^{\pi}f(t)\cos (nt)dt$, where $f\in {C}^1[-\pi,\pi]$ and $f(-\pi)=f(\pi)$. Then,
1) does the sequence $na_n$ converge to $0$ as $n\to\infty$?
2)does the series $\sum_{n=1}^{\infty}n^2|a_n|^2$ converge?
I think yes to both, because of the boundedness of $na_n$. But, how do we prove the convergence? Thanks beforehend.
The answer to the question for part 2) is easy from Parseval's Identity applied to $f'$ as pointed out by AD. The first part follows by integration by parts and the boundedness of $f$, as pointed out in the comments. Note that the second part has a more general version for $f$ possessing only weak derivatives.