Is there a handy way to tell if $\sum_{k=1}^\infty\left(\frac{1}{k}-\frac{1}{2^k}\right)$ diverges or not? I have a hunch that it diverges, since it looks like the sum is just $\zeta(1)-1=\infty$. But I'm not sure one can rearrange the series as $ \sum_{k=1}^\infty\frac{1}{k}-\sum_{k=1}^\infty\frac{1}{2^k}.$
Is that a valid move?