I've recently come across the following series, and I am trying to figure out if there is a closed form solution, or an asymptotic behavior for large M:
$$ \sum_{j=0}^\infty(-1)^j\frac{1 -e^{(aj^2 + b)M}}{1-e^{aj^2 +b}} $$
As far as I know it is not possible to use residues to solve this in the complex plane, because the function doesn't satisfy $|f(z)|<M/|z|^k$. Of the things I've tried none seemed to push me forward!