Let a numerical sequence $\{a_k\}$ tend to zero and be convex. We obtain estimates of $$ g(x) := \sum_{k=1}^{\infty} a_k \sin kx $$ for $x\,\to\,0$ expressed in terms of the coefficients $a_k$. These estimates are of order- or asymptotic character. For example, the following order equality is true: $$ g(x) \sim ma_m + \frac{1}{m} \sum_{k = 1}^{m - 1} k a_k, $$ where $$ x \in łeft ({\frac {\pi}{m+1}, \frac {\pi}{m}} \right ]. $$