Question:

1) How to determine the convergence of

$\displaystyle \sum_{k=1}^{\infty} \frac{\cos(k^{\alpha} x)}{k^{\alpha}} (-1)^k $

where $x \in \mathbb{R}$ and $\alpha \in (0,1]$. I am especially interested in the case of $\alpha = 1/2$.

2) For a fixed $\alpha$, if the above series converges for every $x$, is the convergence uniform? Is the resulting sum bounded in $x$?

I found the series tests (alternating test,etc.) I learned not useful in this situation, except that the convergence is clear for $x = 0$...