There are many different tests for convergence of a series. Are you referring to a specific one? For the specific series you give, it looks to me like either the "ratio test" or "root test" will work. Do you know those?
I'm reviewing Meirovitch's "Methods of Analytical Dynamics," and I don't understand the commutation of the derivative from r to dr:
$$
\mathbf{F} \cdot d\mathbf{r} = m \ddot{\mathbf{r}} \cdot d\mathbf{r} = m\mathbf{\dot{r}} \cdot d\mathbf{\dot{r}}
$$