I have a function $f$ defined on $[-\pi, \pi]$, $f(\pi) = f(-\pi)$, and has continuous first derivative. How can I prove that the sum of the absolute value of the Fourier coefficients converges? The fourier coefficients are defined as $$c_n = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(x) e^{-inx} \, dx$$
I can bound/ find the convergence of $$\sum |c_n|^2 n^2$$ but I don't know what to do from there or if there is an easier method.