Consider the following series where $a, b$ are real numbers $$\sum_{k=1}^{\infty} \frac{(-1)^k}{k} e^{a k + b k^2} $$
Does it admit a closed form or, otherwise "nice", expression?
This form comes up when trying to compute the expectation of $\log(1 + e^{-x})$ where $x$ is normally distributed.
How do I get there?
$$\log(1 + e^{-x}) = -\sum_{k=1}^{\infty}\frac{(-1)^k}{k} e^{-k x}$$
and
$$\int_{-\infty}^{\infty} e^{-k x} \frac{e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}}{\sigma \sqrt{\tau}} = e^{\frac{1}{2}k(k \sigma^2 - 2\mu)}$$
I haven't particularly looked at convergence for now, and with $\sigma^2 >0$ it's obviously not looking good, but I'm trying to manipulate this thing formally and just see what happens, maybe we can land back on our feet somehow...