Positivity and tails of Jacobi theta series

Using elementary $q$-series manipulations, we establish a positivity property for the tails of the Jacobi theta series. Specifically, for integers $k\ge 1$ and $n\ge 0$, define $$\sum_{n\ge0}\sum_{m\in\mathbb{Z}}J_{k,n}(m)z^m q^{n} = \frac{(-1)^k q^{-\binom{k+1}{2}}}{(z)_{\infty}(q/z)_\infty} \sum_{j\ge k}(-1)^jq^{\binom{j+1}{2}}z^{-j}(1-z^{2j+1}),$$ where $(a)_\infty:=\prod_{n\ge0}(1-aq^n)$ denotes the $q$-shifted factorial. We prove that for all integers $k\ge 1$ and $n\ge 0$, the coefficients $J_{k,n}(m)$ are positive for all integers $-(k+n)\le m\le k+n$.