About the convergence of $ \sum_{n=1}^{\infty} a_nx^{1/n}$ 
Let $(a_n)_n \subset \mathbb R$ be a sequence of real numbers and consider the series $$
\sum_{n=1}^{\infty} a_nx^{1/n}.
$$
  Show that if the series converges for some $x_0>0$ then it converges in $(0,+\infty)$.

I was trying to use comparison test at least in the case $x<x_0$ but, as Jonathan's comment pointed out, the $a_n$ may not be positive. 
So the problem is how to start: have you got any ideas? Any hints, please? Thank you.
 A: Hint: Given any $x>0$, note that
$$\sum_{n\geq 1}a_nx^{1/n} = \sum_{n\geq 1}a_nx_0^{1/n}\sqrt[n]{\frac{x}{x_0}}$$
Now, let $b_n = a_nx_0^{1/n}$; $c_n = \sqrt[n]{\frac{x}{x_0}}$.
What tests do you know that would prove the convergence of
$$\sum_{n\geq 1}b_nc_n?$$
Edit (additional hints) Let's recount what we know:


*

*The example of $a_n = \frac{(-1)^n}{n}$ (at, for example, $x_0=1$) shows that the series might converge conditionally (and not absolutely), so any test that would imply absolute convergence is out of the question.

*We know--as you've mentioned in comments--that $\sum_{n\geq 1}b_n$ converges.

*We know that $c_n\to 1$ (hence trivially implying $c_n\not\to 0$, which I only mention because of its importance in some tests). What's more, we know that this convergence is monotonic. We do not, however, know if $(c_n)$ is an increasing/decreasing sequence.


That's about everything of use I can think of, but it should suffice to point you to the convergence test I have in mind (not to imply other profitable approaches don't exist; I just can't think of any).

Edit (answer)

 Abel's convergence test deals with just this sort of situation.

