About a month ago I came across a problem where this particular sum arose: $$\lim\limits_{\delta\to 1}\sum\limits_{k\geq 0}\frac{(-1)^{k}\log(\omega k+\rho)(k+1)}{\delta^{k+2}(\omega k+\rho)},\qquad \omega,\rho\in\mathbb{N}^{+}$$
I wanted to know if there was a way we could evaluate it for some particular values of $\omega$ and $\rho$. The first problem to consider is that of convergence, which only occurs for $|\delta|\gt 1$. However, I think it might be possible to express the sum in terms of derivatives of the Lerch transcendant and then transform it into a sum of derivatives of the Hurwitz Zeta function. I'm particularly interested in evaluating the sum for $(\omega, \rho)\longrightarrow (3,4), (3,3),(3,2)$. But I can do that on my own after figuring out a closed-form expression for the sum. Maybe a renormalization technique could be fruitful here too?
From partial summation and $\sum_{k\le K} (-1)^{k+1}= \frac{1+(-1)^{K+1}}{2}$ we obtain that
$$\lim\limits_{\delta\to 1}\sum\limits_{k\geq 0}\frac{(-1)^{k}\log(\omega k+\rho)(k+1)}{\delta^{k+2}(\omega k+\rho)}= \frac1\omega F'(0)-(1-\frac{\rho}{\omega}) F(1)$$ where $F(s)$ the analytic continuation of $$F(s)=\omega^{-s}\sum_k (-1)^{k+1} ( k+\rho/\omega)^{-s}$$ which is entire.
Then from the infinite product for $\Gamma(s)$ and the same partial summation idea we get that $\exp(F'(0))$ has an expression in term of $\Gamma(\rho/(2\omega))/\Gamma(\rho/(2\omega)+1/2)$ and $F(1)$ has an expression in term of $\frac{\Gamma'}{\Gamma}(\rho/(2\omega))-\frac{\Gamma'}{\Gamma}(\rho/(2\omega)+1/2)$