= Solution
Let $r=e^{i\theta}\ne1$. The <geometric series> from any starting index is bounded:
$$
B_j=\sum_{n=m}^j r^n=\frac{r^m(1-r^{j-m+1})}{1-r},\qquad
|B_j|\leq K=\frac2{|1-r|}.
$$
Using <summation by parts>,
$$
\sum_{n=m}^N a_nr^n=a_NB_N+\sum_{n=m}^{N-1}(a_n-a_{n+1})B_n.
$$
Since the coefficients are nonincreasing and positive, the <modulus> is at most $K(a_N+a_m-a_N)=Ka_m\to0$. The Cauchy criterion proves \b[convergence for every $\theta\notin2\pi\mathbb Z$]. This is a direct proof of the <Dirichlet test> in this setting.
For $a_n=1/n$, the imaginary parts of the convergent partial sums give convergence of the sine series at those angles. If $\theta\in2\pi\mathbb Z$, every sine term is zero. Hence \b[the sine series converges for all real angles].
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