This is my first post here.
I have some troubles with this property of the Fourier coefficients. Indeed, let $f(x)$ be a continuous real function, with compact support $[a,b] \subset (0,2\pi)$, and $ b_k := \frac{1}{\pi}\int_0^{2\pi}f(x)\sin kx \; \mathrm{d}x $
My question is: does the series $ \sum_{k=1}^\infty \frac{b_k}{k}$ converge? If the answer is yes, is its sum equal to $\int_0^{2\pi}\frac{\pi-x}{2}f(x) \; \mathrm{d}x$?
How can I prove/disprove this? What do you suggest? Thanks a lot.