In Serge Lang's complex analysis book there are some theorems that relates convergence with absolute convergence (in the harder direction) ,e.g
If $f(z) = \sum a_nz^n $ has radius of convergence $r>0$, then $f(z)$ is analytic on open disk $D(0, r)$.
Let $f(z)$ be a convergent power series with $a_0 = 0$, and $a_1 \neq 0$, then its inverse $g(z)$ is also absolutely convergent on some disk
And others possibly.Proofs of above uses something similar to below proposition but I am not sure why author did not state it once and use it as given instead use it in similar manners each time, unless it is incorrect. Proposition is about using convergence to show absolute convergence.
I think it would be just convenient to state convergence and then this can imply absolute convergence on some possibly smaller neighborhood.
If $f(z) = \sum a_nz^n$ has radius of convergence $r>0$ then $\sum a_nz^n$ is absolutely convergent on potentially smaller disk around origin.
proof:
We know by convergence of original series, there exists some $A$ such that $|a_n| \leq A^n$ for all n. (This is from the book). Then we if we show that $\sum A^nz^n$ has some non-zero radius of convergence then above follows. Above is geometric series which convergence for $|z| < A^{-1}$.
Anyway, can someone verify above?
As already indicated in the comments, for power series the radius of convergence and radius of absolute convergence coincide. To see this, you may use the limit formula for the radius of convergence for $$ \sum_{n} a_{n}z^{n} $$ and $$ \sum_{n} |a_{n}| |z|^{n} $$ for example.
Your question though has much more to offer in the context of Dirichlet series. That is, series of the form $$\sum_{n=1}^{\infty} \frac{a_{n}}{n^{s}}, \qquad s=\sigma+it. $$ One may prove that such series give rise to holomorphic functions of the complex vairable $s$. Namely, there is an abscissa $\sigma_{c}$ for which the series is convergent for any $\operatorname{Re}(s)>\sigma_{c}$. Note that $\sigma_{c}$ may be infinite, in which case no convergence is possible. This number in some manner plays the role of the radius of convergence in the case of power series.
What differentiates the theory of Dirichlet series from regular power series though, is that abscissas for convergence and absolute convergence may not coincide and one has to move in a smaller half plane in order to achieve this kind of convergence. Not only that, the abscissa of uniform convergence may not coincide with none of the aforementioned, though some relations are established. For example, the Bohnenblust-Hille theorem.
A series exhibiting such behaviour is $$\sum_{n=1}^{\infty} \frac{e^{ib n^{A}}}{n^{s}} $$ for $0<A<1$ and $b>0$. It holds that $$\frac{1}{2}A \leq \sigma_{a}-\sigma_{u}\leq \frac{1}{2}.$$ This may be improved to an equality.
A good introduction in this subject is "The General Theory of Dirichlet's Series" by G.H. Hardy and Marcel Riesz.
I hope this will be helpfull.