I am strugging with the following problem: Show whether the series $$ \sum_{k=1}^\infty k^{-1/2}[\sin (\frac x {2k})-\sin (\frac x {2k-1})] $$ converges uniformly on $\mathbb R$.
It is easy to see that the series converges pointwise, using the following estimate: $$ |\sin (\frac x {2k})-\sin (\frac x {2k-1})|\leq C|x|k^{-2}. $$ But if I attempt to show that the series converges uniformly, I have to use better estimates that deal with large values of $x$. Any suggestions?