Examples of trigonometric series with non-negative partial sums

Let $\{a_n\}_{n=1}^\infty$ be a monotone sequence of non-negative real numbers. In this paper the condition
$$ a_1>0 \quad\text{and}\quad \sum _{k=1}^n(-1)^{k-1}ka_k\geqslant 0 \quad\text{for all $n\geqslant 1$} $$
are proved to be necessary and sufficient for all partial sums of the trigonometric sine series $\sum _{n=1}^\infty a_n\sin(nx)$ to be positive on the interval $(0,\pi)$. New conditions on the coefficients of a trigonometric cosine series ensuring that all its partial sums are positive on the real axis are presented.