Let $0<\alpha <1$ and $\beta > 0$. Consider the mapping $$F(\alpha, \beta) = \sum_{n=1}^{\infty}{\dfrac{(-1)^{n-1}\biggl( \cos \left(\beta\ln(n)\right)\biggr)}{n^{\alpha}}}.$$ Can we prove $F(\alpha, \beta)$ exist (in other words the series converges), and how should one approach studying the roots of $F(\alpha, \beta)$ that is $$F(\alpha, \beta) = 0.$$ Numerical results suggest the partial sums of this series satisfies $$F_k(\alpha,\beta)\le F(\alpha, \beta)\le F_{k+1}(\alpha,\beta)$$ for $k$ provided sufficently large. I want to then look for a $C^k$ (infinitely differential in $k$) function $g(k,\alpha,\beta)$ such that  $$\dfrac{\partial }{\partial k} g(k,\alpha, \beta) = 0 $$ and $$g(k,\alpha, \beta) = F_k(\alpha, \beta)$$ for each $k\in \mathbb{N}$ (but not necessarily all real numbers) so that at each point of  $k$ the partial sum the derivative is zero and we get a maximum or minimum (wave like structure) of $g$. Can we find such a function $\textbf{g}$? If so, I claim $g$ will converge to $F(\alpha,\beta)$ for large $k$. That is $$\lim_{k\rightarrow \infty}{g(k,\alpha,\beta)} = F(\alpha,\beta)$$ but can we prove it?