Does these sequence and series converge? Let $f\in C^1[-\pi,\pi]$ st $f(-\pi)=f(\pi)$ and define $$a_n=\int^{\pi}_{-\pi} f(t)\cos nt dt\,$$ for $n \in\Bbb{N}$ . Then does the sequence $\{na_n\}$ converges? And does the series $\sum^{\infty}_{n=1} n^2|a_n|^2$ converges as $n\to \infty$.
I know that $a_n \to 0$ as $n\to \infty$ and if the series converges the $\{na_n\}$ must converge to $0$.

By integration by parts, we have:
$$ n\, a_n = \int_{-\pi}^{\pi} f(t)\, n\cos(nt)\, dt = -\int_{-\pi}^{\pi} f'(t)\sin(n t)\,dt \tag{1}$$
and since $f'\in C^0([-\pi,\pi])$, the RHS of $(1)$ converges to zero by the Riemann-Lebesgue lemma.
$f'\in C^0([-\pi,\pi])$ implies $f'\in L^2([-\pi,\pi])$, hence
$$ \sum_{n\geq 1} n^2 a_n^2 <+\infty $$
follows from Bessel's inequality.
