I want to show that \begin{equation} \sum_{\substack{k \geq 1 \\ k \text{ odd}}} k e^{-k^2 a} \sin(kx) \geq 0 \qquad \text{for all } x \in [0,\pi], \, a > 0. \end{equation}
How should I start? I tried approximating the infinite sum by its partial sums, but the terms oscillate and the number of terms needed for a good enough approximation increases without bound as $a \to 0$. Perhaps somebody can recognize this function as the Fourier series (?) of some better-known function, or provide some inequalities for dealing with such series that I am not aware of.