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Let $\sum_{k=1}^\infty a_k$ be a convergent series. Then can we obtain $\sum_{k=1}^\infty a_k\sin (k\pi x)$ converges for $x$ irrational?
If $\sum_{k=1}^\infty a_k$ converges absolutely, then I can obtain that $\sum_{k=1}^\infty a_k\sin (k\pi x)$ converges for all $x$. But I do not know how to deal with the case $\sum_{k=1}^\infty a_k$ converges conditionally...
From this proof, $\sum_{k>0} \sin(k)$ is bounded, and from Dirichlet's test, $\sum_{k>0} \sin(k)/k$ converges conditionally. However, for $x=1/\pi$, $\sum_{k>0} \sin(k\pi x)\sin(k)/k > \sum_{k' \in C} \alpha/k'$ where $0<\alpha<1$ and $C$ is a subsample of $\mathbb{N}$ such that $card(C \cup \{1,...N\})\geq \beta N$, $\beta>0$ depends on $\alpha$. Thus $\sum_{k' \in C} 1/k'$ diverges.
