I need to classify the following series:

$\displaystyle \sum_{n=n_0}^\infty\sqrt{a_n}$

with $\displaystyle (a_n)\ge0$ and $\displaystyle \sum_{n=n_0}^\infty a_n$ $\displaystyle \mathbb{C}$

I guess the question on the whole is whether i can classify any of the following series:

$\displaystyle \sum_{n=n_0}^\infty (a_n)^r$ , $\displaystyle \forall r\in\mathbb{Z}$

based on $\displaystyle \sum_{n=n_0}^\infty a_n$ being convergent

As far as I can tell (using the Comparison Test), $\displaystyle \sum_{n=n_0}^\infty (a_n)^r$ will converge if $\displaystyle r > 1$, but what if $\displaystyle r < 1$?

Thanks in advance.