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Dirichlet Test: Let $(a_n)$ and $(b_n)$ be two sequences satisfying $\lim_{n\to+\infty}a_n=0$, $\sum_{n=1}^{+\infty}|a_{n+1}-a_{n}|$ converges, and There exists a number $M$ such that $\bigl|\sum_{n=1}^m b_n\bigr|\leq M$ for all $m\geq 1$. Then the sum $\sum_{n=1}^{+\infty} a_nb_n$ converges. Let us apply that to the sum ...
We can use the Abel summation and get$$\sum_{n\leq N}\frac{\sin\left(nx\right)}{n}=H_{N}\sin\left(Nx\right)-x\int_{1}^{N}H_{\left\lfloor t\right\rfloor }\cos\left(tx\right)dt$$ and now using the asymptotic $$H_{n}=\log\left(n\right)+\gamma+O\left(\frac{1}{n}\right)$$ we have \sum_{n\leq ...