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 $$\sum _{n\ge 0}\sum _{m\in \mathbb{Z}}{J}_{k,n}(m)z^m{q}^n=\frac{(-1{)}^k{q}^{-\left(\genfrac{}{}{0px}{}{k+1}{2}\right)}}{(z{)}_{\mathrm{\infty }}(q\mathrm{/}z{)}_{\mathrm{\infty }}}\sum _{j\ge k}(-1{)}^j{q}^{\left(\genfrac{}{}{0px}{}{j+1}{2}\right)}{z}^{-j}(1-z^{2j+1}),$$ where $(a{)}_{\mathrm{\infty }}:={\prod }_{n\ge 0}(1-a{q}^n)$ 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 $-(k+n)\le m\le k+n$.