Series $\sum\frac{1}{b_n}$ where $ b_n=\sum_{k=1}^n(-1)^{k+1}\sqrt{k}.$ Let 
$$b_n=\sum_{k=1}^n(-1)^{k+1}\sqrt{k}.$$
I look for the following series $\displaystyle \sum\frac{1}{b_n}$ is convergent or divergent ? Any suggestion ?

The series $b_n$ is not convergent, so $\sum_{k=1}^\infty \frac{1}{b_n}$ is probably convergent. Though you would have to use some sort of test, maybe ratio or integral or something of the sort.
EDIT: I interpreted $b_n$ with $\sqrt k$ and not $\sqrt n$, as I'm not totally sure what you're writing.
