ca.analysis and odes - convergence of a series involving cosines

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 $x = 0$...