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...