I am considering if the Fourier series development of $f(\theta)$ is \begin{equation} f(\theta) = \sum_{n=1}^\infty a_n \cos(n\theta) + b_n \sin(n\theta) \end{equation}
Is there any relationship between the series \begin{equation} \sum_{n=1}^\infty \dfrac{a_n^2 + b_n^2}{n} \end{equation} and $f(\theta)$?