convergence of $\sum\limits_{n=1}^\infty \frac{(-1)^{\lfloor \sqrt{n}\rfloor}}{\sqrt{n}}$ Does anybody know how to show analytically the convergence (divergence ?) of $\enspace\displaystyle \sum\limits_{n=1}^\infty \frac{(-1)^{\lfloor \sqrt{n}\rfloor}}{\sqrt{n}}\enspace$ or some literature for it ? (Thanks!) 

Note: Unfortunately I am still missing a useful approach. 

EDIT:

I want to thank all here for the nice help. 

Result: The series is divergent (but limited) as it is written. 

It's possible to change to $\enspace\displaystyle\sum\limits_{n=1}^\infty (-1)^n (-2+\sum\limits_{k=n^2}^{(n+1)^2-1}\frac{1}{\sqrt{k}})\enspace$ to avoid the oscillation and to get convergence.
You can prove that it diverges by observing that the sum of each "block" is greater than a constant, that is, $$\sum_{k=n^2}^{(n+1)^2-1}\frac{1}{\sqrt{k}}&gt;C$$for some positive $C$.
It's alternating series (with periodically strange alternation, but alternation nonetheless) and thus converges according to alternating series test.