if $c_nr^n$ is bounded for $r \geq 0$, then for $0<r'<r$ we have that $\sum_{n=0}^\infty |c_n r'^n| =\sum_{n=0}^\infty |c_n r^n| \left(\frac{r'}{r}\right)^n \leq \sup_{n \in \Bbb N}(c_nr^n)\cdot \sum_{n=0}^\infty\left(\frac{r'}{r}\right)^n<\infty$. So $\sum_{n=0}^\infty c_n r'^n$ converges absolutely
On the other hand, if $\sum_{n =0}^\infty c_n x^n$ converges, then $\lim_{n \to \infty} c_n x^n =0$, so $c_nx^n$ is bounded and hence also $c_n|x|^n$ is bounded.
These two arguments give you two inequalities between the two definitions.