I've been trying to figure this out for days now, but I have no idea how to show this. It's from Partial Differential Equations: An Introduction by Walter A. Strauss.
Suppose $\int_{-\pi}^{\pi} [ |f(x)|^2 + |g(x)|^2 ] dx $ is finite where $g(x) = \frac{f(x)}{e^{ix} - 1}$. Let $c_n$ be the coefficients of the complex Fourier series of $f(x)$. Show that $\sum_{-N}^N c_n \rightarrow 0$ as $N \rightarrow 0$.
Let $f(x) = \sum c_n e^{inx}$ and $g(x) = \sum d_n e^{inx}$. Since $\int_{-\pi}^\pi |f(x)|^2 \, dx < \infty$, and $\int_{-\pi}^\pi |g(x)|^2 \, dx < \infty$, by Parseval's identity, we know that $\sum |c_n|^2 < \infty$ and $\sum |d_n|^2 < \infty$.
Also, since $f(x) = g(x)(e^{ix}-1)$, we have $c_n = d_{n-1} - d_{n}$. Hence $$ d_{-n} = d_0 - c_0 - c_{-1} - \dots - c_{-n+1} \text{ for $n \ge 0$.}$$ Since $d_{-n}\to 0$ as $n \to \infty$, it follows that $$ d_0 = \sum_{k=0}^\infty c_{-k} .$$ Also $$ d_n = c_n + c_{n-1} + \dots + c_{1} + d_0 .$$ Since $d_{n}\to 0$ as $n \to \infty$, it follows that $$ d_0 + \sum_{k=1}^{\infty} c_k = 0 .$$
