Given a double sum \begin{equation} \sum\limits_{n=1}^{\infty} (-1)^{n-1} \sum\limits_{i=1}^{n} \dfrac{1}{(n+1-i)^{\beta}i^{\alpha}} \end{equation} Depending on parameters $\alpha, \beta$ find out when does this series convrge and diverge. It is also known that both $\alpha$ and $\beta$ are positive numbers.
This problem, however, arose from another, more typical one, I'll describe it, too:
Given two series: \begin{equation} \sum\limits_{n=1}^{\infty} \dfrac{(-1)^{n-1}}{n^{\alpha}}, \sum\limits_{n=1}^{\infty} \dfrac{(-1)^{n-1}}{n^{\beta}} \end{equation} where $(\alpha,\beta)\not = 0$. Prove that the multiplication product of those two series converges when $\alpha+\beta > 1$, and diverges when $\alpha+\beta < 1$.
What I really did here is that I've just wrote how the multiplication product elements would look like, and what I've got you can see in the first formula. Afterwards I've figured out I can solve this problem using Mertens theorem, it immediately gives us the fact that when $\alpha+\beta > 1$ the multiplication product converges, but I still have no idea how to work through the case $\alpha+\beta = 1$. Also I got really interested if there is any way to work with this double sum and extract the right answer from it.